Calculus Examples

$\frac{4 f^{2} P}{1 - f^{2}} = K$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d P} (\frac{4 f^{2} P}{1 - f^{2}}) = \frac{d}{d P} (K)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{d}{d P} [\frac{4 f^{2} P}{1^{2} - f^{2}}]$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = f$ .

$\frac{d}{d P} [\frac{4 f^{2} P}{(1 + f) (1 - f)}]$

Step 2.2

Since $4$ is constant with respect to $P$ , the derivative of $\frac{4 f^{2} P}{(1 + f) (1 - f)}$ with respect to $P$ is $4 \frac{d}{d P} [\frac{f^{2} P}{(1 + f) (1 - f)}]$ .

$4 \frac{d}{d P} [\frac{f^{2} P}{(1 + f) (1 - f)}]$

Step 2.3

Differentiate using the Quotient Rule which states that $\frac{d}{d P} [\frac{f (P)}{g (P)}]$ is $\frac{g (P) \frac{d}{d P} [f (P)] - f (P) \frac{d}{d P} [g (P)]}{{g (P)}^{2}}$ where $f (P) = f^{2} P$ and $g (P) = (1 + f) (1 - f)$ .

$4 \frac{(1 + f) (1 - f) \frac{d}{d P} [f^{2} P] - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.4

Differentiate using the Product Rule which states that $\frac{d}{d P} [f (P) g (P)]$ is $f (P) \frac{d}{d P} [g (P)] + g (P) \frac{d}{d P} [f (P)]$ where $f (P) = f^{2}$ and $g (P) = P$ .

$4 \frac{(1 + f) (1 - f) (f^{2} \frac{d}{d P} [P] + P \frac{d}{d P} [f^{2}]) - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.5

Differentiate using the Power Rule.

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Step 2.5.1

Differentiate using the Power Rule which states that $\frac{d}{d P} [P^{n}]$ is $n P^{n - 1}$ where $n = 1$ .

$4 \frac{(1 + f) (1 - f) (f^{2} \cdot 1 + P \frac{d}{d P} [f^{2}]) - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.5.2

Multiply $f^{2}$ by $1$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + P \frac{d}{d P} [f^{2}]) - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d P} [f (g (P))]$ is $f' (g (P)) g' (P)$ where $f (P) = P^{2}$ and $g (P) = f$ .

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Step 2.6.1

To apply the Chain Rule, set $u$ as $f$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + P (\frac{d}{d u} [u^{2}] \frac{d}{d P} [f])) - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + P (2 u \frac{d}{d P} [f])) - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.6.3

Replace all occurrences of $u$ with $f$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + P (2 f \frac{d}{d P} [f])) - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.7

Move $2$ to the left of $P$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 \cdot P f \frac{d}{d P} [f]) - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.8

Rewrite $\frac{d}{d P} [f]$ as $f'$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P \frac{d}{d P} [(1 + f) (1 - f)]}{{((1 + f) (1 - f))}^{2}}$

Step 2.9

Differentiate using the Product Rule which states that $\frac{d}{d P} [f (P) g (P)]$ is $f (P) \frac{d}{d P} [g (P)] + g (P) \frac{d}{d P} [f (P)]$ where $f (P) = 1 + f$ and $g (P) = 1 - f$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) \frac{d}{d P} [1 - f] + (1 - f) \frac{d}{d P} [1 + f])}{{((1 + f) (1 - f))}^{2}}$

Step 2.10

Differentiate.

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Step 2.10.1

By the Sum Rule, the derivative of $1 - f$ with respect to $P$ is $\frac{d}{d P} [1] + \frac{d}{d P} [- f]$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (\frac{d}{d P} [1] + \frac{d}{d P} [- f]) + (1 - f) \frac{d}{d P} [1 + f])}{{((1 + f) (1 - f))}^{2}}$

Step 2.10.2

Since $1$ is constant with respect to $P$ , the derivative of $1$ with respect to $P$ is $0$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (0 + \frac{d}{d P} [- f]) + (1 - f) \frac{d}{d P} [1 + f])}{{((1 + f) (1 - f))}^{2}}$

Step 2.10.3

Add $0$ and $\frac{d}{d P} [- f]$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) \frac{d}{d P} [- f] + (1 - f) \frac{d}{d P} [1 + f])}{{((1 + f) (1 - f))}^{2}}$

Step 2.10.4

Since $- 1$ is constant with respect to $P$ , the derivative of $- f$ with respect to $P$ is $- \frac{d}{d P} [f]$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- \frac{d}{d P} [f]) + (1 - f) \frac{d}{d P} [1 + f])}{{((1 + f) (1 - f))}^{2}}$

Step 2.11

Rewrite $\frac{d}{d P} [f]$ as $f'$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) \frac{d}{d P} [1 + f])}{{((1 + f) (1 - f))}^{2}}$

Step 2.12

By the Sum Rule, the derivative of $1 + f$ with respect to $P$ is $\frac{d}{d P} [1] + \frac{d}{d P} [f]$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) (\frac{d}{d P} [1] + \frac{d}{d P} [f]))}{{((1 + f) (1 - f))}^{2}}$

Step 2.13

Since $1$ is constant with respect to $P$ , the derivative of $1$ with respect to $P$ is $0$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) (0 + \frac{d}{d P} [f]))}{{((1 + f) (1 - f))}^{2}}$

Step 2.14

Add $0$ and $\frac{d}{d P} [f]$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) \frac{d}{d P} [f])}{{((1 + f) (1 - f))}^{2}}$

Step 2.15

Rewrite $\frac{d}{d P} [f]$ as $f'$ .

$4 \frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) f')}{{((1 + f) (1 - f))}^{2}}$

Step 2.16

Combine $4$ and $\frac{(1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) f')}{{((1 + f) (1 - f))}^{2}}$ .

$\frac{4 ((1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) f'))}{{((1 + f) (1 - f))}^{2}}$

Step 2.17

Simplify.

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Step 2.17.1

Apply the product rule to $(1 + f) (1 - f)$ .

$\frac{4 ((1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P ((1 + f) (- f') + (1 - f) f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.2

Apply the distributive property.

$\frac{4 ((1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P (1 (- f') + f (- f') + (1 - f) f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.3

Apply the distributive property.

$\frac{4 ((1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P (1 (- f') + f (- f') + 1 f' - f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.4

Apply the distributive property.

$\frac{4 ((1 + f) (1 - f) (f^{2} + 2 P f f') - f^{2} P (1 (- f')) - f^{2} P (f (- f')) - f^{2} P (1 f') - f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.5

Apply the distributive property.

$\frac{4 ((1 + f) (1 - f) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6

Simplify the numerator.

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Step 2.17.6.1

Simplify each term.

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Step 2.17.6.1.1

Expand $(1 + f) (1 - f)$ using the FOIL Method.

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Step 2.17.6.1.1.1

Apply the distributive property.

$\frac{4 ((1 (1 - f) + f (1 - f)) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.1.2

Apply the distributive property.

$\frac{4 ((1 \cdot 1 + 1 (- f) + f (1 - f)) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.1.3

Apply the distributive property.

$\frac{4 ((1 \cdot 1 + 1 (- f) + f \cdot 1 + f (- f)) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2

Simplify and combine like terms.

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Step 2.17.6.1.2.1

Simplify each term.

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Step 2.17.6.1.2.1.1

Multiply $1$ by $1$ .

$\frac{4 ((1 + 1 (- f) + f \cdot 1 + f (- f)) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2.1.2

Multiply $- f$ by $1$ .

$\frac{4 ((1 - f + f \cdot 1 + f (- f)) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2.1.3

Multiply $f$ by $1$ .

$\frac{4 ((1 - f + f + f (- f)) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{4 ((1 - f + f - f \cdot f) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2.1.5

Multiply $f$ by $f$ by adding the exponents.

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Step 2.17.6.1.2.1.5.1

Move $f$ .

$\frac{4 ((1 - f + f - (f \cdot f)) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2.1.5.2

Multiply $f$ by $f$ .

$\frac{4 ((1 - f + f - f^{2}) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2.2

Add $- f$ and $f$ .

$\frac{4 ((1 + 0 - f^{2}) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.2.3

Add $1$ and $0$ .

$\frac{4 ((1 - f^{2}) (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.3

Expand $(1 - f^{2}) (f^{2} + 2 P f f')$ using the FOIL Method.

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Step 2.17.6.1.3.1

Apply the distributive property.

$\frac{4 (1 (f^{2} + 2 P f f') - f^{2} (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.3.2

Apply the distributive property.

$\frac{4 (1 f^{2} + 1 (2 P f f') - f^{2} (f^{2} + 2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.3.3

Apply the distributive property.

$\frac{4 (1 f^{2} + 1 (2 P f f') - f^{2} f^{2} - f^{2} (2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4

Simplify each term.

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Step 2.17.6.1.4.1

Multiply $f^{2}$ by $1$ .

$\frac{4 (f^{2} + 1 (2 P f f') - f^{2} f^{2} - f^{2} (2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.2

Multiply $2 P f f'$ by $1$ .

$\frac{4 (f^{2} + 2 P f f' - f^{2} f^{2} - f^{2} (2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.3

Multiply $f^{2}$ by $f^{2}$ by adding the exponents.

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Step 2.17.6.1.4.3.1

Move $f^{2}$ .

$\frac{4 (f^{2} + 2 P f f' - (f^{2} f^{2}) - f^{2} (2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 (f^{2} + 2 P f f' - f^{2 + 2} - f^{2} (2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.3.3

Add $2$ and $2$ .

$\frac{4 (f^{2} + 2 P f f' - f^{4} - f^{2} (2 P f f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.4

Multiply $f^{2}$ by $f$ by adding the exponents.

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Step 2.17.6.1.4.4.1

Move $f$ .

$\frac{4 (f^{2} + 2 P f f' - f^{4} - (f \cdot f^{2}) (2 P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.4.2

Multiply $f$ by $f^{2}$ .

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Step 2.17.6.1.4.4.2.1

Raise $f$ to the power of $1$ .

$\frac{4 (f^{2} + 2 P f f' - f^{4} - (f^{1} f^{2}) (2 P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 (f^{2} + 2 P f f' - f^{4} - f^{1 + 2} (2 P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.4.3

Add $1$ and $2$ .

$\frac{4 (f^{2} + 2 P f f' - f^{4} - f^{3} (2 P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.4.5

Multiply $2$ by $- 1$ .

$\frac{4 (f^{2} + 2 P f f' - f^{4} - 2 f^{3} (P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.5

Apply the distributive property.

$\frac{4 f^{2} + 4 (2 P f f') + 4 (- f^{4}) + 4 (- 2 f^{3} (P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.6

Simplify.

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Step 2.17.6.1.6.1

Multiply $2$ by $4$ .

$\frac{4 f^{2} + 8 (P f f') + 4 (- f^{4}) + 4 (- 2 f^{3} (P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.6.2

Multiply $- 1$ by $4$ .

$\frac{4 f^{2} + 8 (P f f') - 4 f^{4} + 4 (- 2 f^{3} (P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.6.3

Multiply $- 2$ by $4$ .

$\frac{4 f^{2} + 8 (P f f') - 4 f^{4} - 8 (f^{3} (P f')) + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.7

Remove parentheses.

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 (- f^{2} P (1 (- f'))) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.8

Multiply $- f'$ by $1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 (- f^{2} P (- f')) + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.9

Multiply $- f^{2} P (- f')$ .

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Step 2.17.6.1.9.1

Multiply $- 1$ by $- 1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 (1 f^{2} P f') + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.9.2

Multiply $f^{2}$ by $1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 (f^{2} P f') + 4 (- f^{2} P (f (- f'))) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.10

Multiply $f^{2}$ by $f$ by adding the exponents.

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Step 2.17.6.1.10.1

Move $f$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 (- (f \cdot f^{2}) P (- f')) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.10.2

Multiply $f$ by $f^{2}$ .

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Step 2.17.6.1.10.2.1

Raise $f$ to the power of $1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 (- (f^{1} f^{2}) P (- f')) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 (- f^{1 + 2} P (- f')) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.10.3

Add $1$ and $2$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 (- f^{3} P (- f')) + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.11

Multiply $- f^{3} P (- f')$ .

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Step 2.17.6.1.11.1

Multiply $- 1$ by $- 1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 (1 f^{3} P f') + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.11.2

Multiply $f^{3}$ by $1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 (f^{3} P f') + 4 (- f^{2} P (1 f')) + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.12

Multiply $f'$ by $1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' + 4 (- f^{2} P f') + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.13

Multiply $- 1$ by $4$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 (f^{2} P f') + 4 (- f^{2} P (- f f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.14

Multiply $f^{2}$ by $f$ by adding the exponents.

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Step 2.17.6.1.14.1

Move $f$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 (- (f \cdot f^{2}) P (- 1 f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.14.2

Multiply $f$ by $f^{2}$ .

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Step 2.17.6.1.14.2.1

Raise $f$ to the power of $1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 (- (f^{1} f^{2}) P (- 1 f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.14.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 (- f^{1 + 2} P (- 1 f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.14.3

Add $1$ and $2$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 (- f^{3} P (- 1 f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.15

Rewrite $- 1 f'$ as $- f'$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 (- f^{3} P (- f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.16

Multiply $- f^{3} P (- f')$ .

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Step 2.17.6.1.16.1

Multiply $- 1$ by $- 1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 (1 f^{3} P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.1.16.2

Multiply $f^{3}$ by $1$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 (f^{3} P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 f^{3} P f'}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.2

Combine the opposite terms in $4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{2} P f' + 4 f^{3} P f' - 4 f^{2} P f' + 4 f^{3} P f'$ .

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Step 2.17.6.2.1

Subtract $4 f^{2} P f'$ from $4 f^{2} P f'$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{3} P f' + 0 + 4 f^{3} P f'}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.2.2

Add $4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{3} P f'$ and $0$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 8 f^{3} (P f') + 4 f^{3} P f' + 4 f^{3} P f'}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.3

Add $- 8 f^{3} P f'$ and $4 f^{3} P f'$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} - 4 f^{3} P f' + 4 f^{3} P f'}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.4

Combine the opposite terms in $4 f^{2} + 8 P f f' - 4 f^{4} - 4 f^{3} P f' + 4 f^{3} P f'$ .

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Step 2.17.6.4.1

Add $- 4 f^{3} P f'$ and $4 f^{3} P f'$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4} + 0}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.6.4.2

Add $4 f^{2} + 8 P f f' - 4 f^{4}$ and $0$ .

$\frac{4 f^{2} + 8 P f f' - 4 f^{4}}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.7

Reorder terms.

$\frac{- 4 f^{4} + 4 f^{2} + 8 P f f'}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.8

Factor $4 f$ out of $- 4 f^{4} + 4 f^{2} + 8 P f f'$ .

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Step 2.17.8.1

Factor $4 f$ out of $- 4 f^{4}$ .

$\frac{4 f (- f^{3}) + 4 f^{2} + 8 P f f'}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.8.2

Factor $4 f$ out of $4 f^{2}$ .

$\frac{4 f (- f^{3}) + 4 f (f) + 8 P f f'}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.8.3

Factor $4 f$ out of $8 P f f'$ .

$\frac{4 f (- f^{3}) + 4 f (f) + 4 f (2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.8.4

Factor $4 f$ out of $4 f (- f^{3}) + 4 f (f)$ .

$\frac{4 f (- f^{3} + f) + 4 f (2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.8.5

Factor $4 f$ out of $4 f (- f^{3} + f) + 4 f (2 P f')$ .

$\frac{4 f (- f^{3} + f + 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.9

Factor $- 1$ out of $- f^{3}$ .

$\frac{4 f (- (f^{3}) + f + 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.10

Factor $- 1$ out of $f$ .

$\frac{4 f (- (f^{3}) - 1 (- f) + 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.11

Factor $- 1$ out of $- (f^{3}) - 1 (- f)$ .

$\frac{4 f (- (f^{3} - f) + 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.12

Factor $- 1$ out of $2 P f'$ .

$\frac{4 f (- (f^{3} - f) - (- 2 P f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.13

Factor $- 1$ out of $- (f^{3} - f) - (- 2 P f')$ .

$\frac{4 f (- (f^{3} - f - 2 P f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.14

Rewrite $- (f^{3} - f - 2 P f')$ as $- 1 (f^{3} - f - 2 P f')$ .

$\frac{4 f (- 1 (f^{3} - f - 2 P f'))}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.15

Move the negative in front of the fraction.

$- \frac{(4 f) (f^{3} - f - 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 2.17.16

Reorder factors in $- \frac{(4 f) (f^{3} - f - 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$ .

$- \frac{4 f (f^{3} - f - 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}}$

Step 3

Since $K$ is constant with respect to $P$ , the derivative of $K$ with respect to $P$ is $0$ .

$0$

Step 4

Reform the equation by setting the left side equal to the right side.

$- \frac{4 f (f^{3} - f - 2 P f')}{{(1 + f)}^{2} {(1 - f)}^{2}} = 0$

Step 5

Solve for $f'$ .

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Step 5.1

Set the numerator equal to zero.

$4 f (f^{3} - f - 2 P f') = 0$

Step 5.2

Solve the equation for $f'$ .

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Step 5.2.1

Divide each term in $4 f (f^{3} - f - 2 P f') = 0$ by $4 f$ and simplify.

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Step 5.2.1.1

Divide each term in $4 f (f^{3} - f - 2 P f') = 0$ by $4 f$ .

$\frac{4 f (f^{3} - f - 2 P f')}{4 f} = \frac{0}{4 f}$

Step 5.2.1.2

Simplify the left side.

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Step 5.2.1.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.2.1.1

Cancel the common factor.

$\frac{4 f (f^{3} - f - 2 P f')}{4 f} = \frac{0}{4 f}$

Step 5.2.1.2.1.2

Rewrite the expression.

$\frac{f (f^{3} - f - 2 P f')}{f} = \frac{0}{4 f}$

Step 5.2.1.2.2

Cancel the common factor of $f$ .

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Step 5.2.1.2.2.1

Cancel the common factor.

$\frac{f (f^{3} - f - 2 P f')}{f} = \frac{0}{4 f}$

Step 5.2.1.2.2.2

Divide $f^{3} - f - 2 P f'$ by $1$ .

$f^{3} - f - 2 P f' = \frac{0}{4 f}$

Step 5.2.1.3

Simplify the right side.

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Step 5.2.1.3.1

Cancel the common factor of $0$ and $4$ .

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Step 5.2.1.3.1.1

Factor $4$ out of $0$ .

$f^{3} - f - 2 P f' = \frac{4 (0)}{4 f}$

Step 5.2.1.3.1.2

Cancel the common factors.

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Step 5.2.1.3.1.2.1

Factor $4$ out of $4 f$ .

$f^{3} - f - 2 P f' = \frac{4 (0)}{4 (f)}$

Step 5.2.1.3.1.2.2

Cancel the common factor.

$f^{3} - f - 2 P f' = \frac{4 \cdot 0}{4 f}$

Step 5.2.1.3.1.2.3

Rewrite the expression.

$f^{3} - f - 2 P f' = \frac{0}{f}$

Step 5.2.1.3.2

Divide $0$ by $f$ .

$f^{3} - f - 2 P f' = 0$

Step 5.2.2

Move all terms not containing $f'$ to the right side of the equation.

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Step 5.2.2.1

Subtract $f^{3}$ from both sides of the equation.

$- f - 2 P f' = - f^{3}$

Step 5.2.2.2

Add $f$ to both sides of the equation.

$- 2 P f' = - f^{3} + f$

Step 5.2.3

Divide each term in $- 2 P f' = - f^{3} + f$ by $- 2 P$ and simplify.

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Step 5.2.3.1

Divide each term in $- 2 P f' = - f^{3} + f$ by $- 2 P$ .

$\frac{- 2 P f'}{- 2 P} = \frac{- f^{3}}{- 2 P} + \frac{f}{- 2 P}$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{- 2 P f'}{- 2 P} = \frac{- f^{3}}{- 2 P} + \frac{f}{- 2 P}$

Step 5.2.3.2.1.2

Rewrite the expression.

$\frac{P f'}{P} = \frac{- f^{3}}{- 2 P} + \frac{f}{- 2 P}$

Step 5.2.3.2.2

Cancel the common factor of $P$ .

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Step 5.2.3.2.2.1

Cancel the common factor.

$\frac{P f'}{P} = \frac{- f^{3}}{- 2 P} + \frac{f}{- 2 P}$

Step 5.2.3.2.2.2

Divide $f'$ by $1$ .

$f' = \frac{- f^{3}}{- 2 P} + \frac{f}{- 2 P}$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Simplify each term.

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Step 5.2.3.3.1.1

Dividing two negative values results in a positive value.

$f' = \frac{f^{3}}{2 P} + \frac{f}{- 2 P}$

Step 5.2.3.3.1.2

Move the negative in front of the fraction.

$f' = \frac{f^{3}}{2 P} - \frac{f}{2 P}$

Step 6

Replace $f'$ with $\frac{d f}{d P}$ .

$\frac{d f}{d P} = \frac{f^{3}}{2 P} - \frac{f}{2 P}$