Calculus Examples

$\frac{d π}{d t} = k P (L - P) (P - k)$

Step 1

Reduce the expression $\frac{d π}{d t}$ by cancelling the common factors.

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Step 1.1

Cancel the common factor.

$\frac{d π}{d t} = k P (L - P) (P - k)$

Step 1.2

Rewrite the expression.

$\frac{π}{t} = k P (L - P) (P - k)$

Step 2

Solve the equation.

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Step 2.1

Find the LCD of the terms in the equation.

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Step 2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$t, 1$

Step 2.1.2

The LCM of one and any expression is the expression.

$t$

Step 2.2

Multiply each term in $\frac{π}{t} = k P (L - P) (P - k)$ by $t$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $\frac{π}{t} = k P (L - P) (P - k)$ by $t$ .

$\frac{π}{t} t = k P (L - P) (P - k) t$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $t$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{π}{t} t = k P (L - P) (P - k) t$

Step 2.2.2.1.2

Rewrite the expression.

$π = k P (L - P) (P - k) t$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Apply the distributive property.

$π = (k P L + k P (- P)) (P - k) t$

Step 2.2.3.2

Rewrite using the commutative property of multiplication.

$π = (k P L - k P \cdot P) (P - k) t$

Step 2.2.3.3

Multiply $P$ by $P$ by adding the exponents.

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Step 2.2.3.3.1

Move $P$ .

$π = (k P L - k (P \cdot P)) (P - k) t$

Step 2.2.3.3.2

Multiply $P$ by $P$ .

$π = (k P L - k P^{2}) (P - k) t$

Step 2.2.3.4

Expand $(k P L - k P^{2}) (P - k)$ using the FOIL Method.

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Step 2.2.3.4.1

Apply the distributive property.

$π = (k P L (P - k) - k P^{2} (P - k)) t$

Step 2.2.3.4.2

Apply the distributive property.

$π = (k P L P + k P L (- k) - k P^{2} (P - k)) t$

Step 2.2.3.4.3

Apply the distributive property.

$π = (k P L P + k P L (- k) - k P^{2} P - k P^{2} (- k)) t$

Step 2.2.3.5

Simplify terms.

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Step 2.2.3.5.1

Simplify each term.

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Step 2.2.3.5.1.1

Multiply $P$ by $P$ by adding the exponents.

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Step 2.2.3.5.1.1.1

Move $P$ .

$π = (k (P \cdot P) L + k P L (- k) - k P^{2} P - k P^{2} (- k)) t$

Step 2.2.3.5.1.1.2

Multiply $P$ by $P$ .

$π = (k P^{2} L + k P L (- k) - k P^{2} P - k P^{2} (- k)) t$

Step 2.2.3.5.1.2

Rewrite using the commutative property of multiplication.

$π = (k P^{2} L - k P L k - k P^{2} P - k P^{2} (- k)) t$

Step 2.2.3.5.1.3

Multiply $k$ by $k$ by adding the exponents.

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Step 2.2.3.5.1.3.1

Move $k$ .

$π = (k P^{2} L - (k \cdot k) P L - k P^{2} P - k P^{2} (- k)) t$

Step 2.2.3.5.1.3.2

Multiply $k$ by $k$ .

$π = (k P^{2} L - k^{2} P L - k P^{2} P - k P^{2} (- k)) t$

Step 2.2.3.5.1.4

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

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Step 2.2.3.5.1.4.1

Move $P$ .

$π = (k P^{2} L - k^{2} P L - k (P \cdot P^{2}) - k P^{2} (- k)) t$

Step 2.2.3.5.1.4.2

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

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Step 2.2.3.5.1.4.2.1

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

$π = (k P^{2} L - k^{2} P L - k (P^{1} P^{2}) - k P^{2} (- k)) t$

Step 2.2.3.5.1.4.2.2

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

$π = (k P^{2} L - k^{2} P L - k P^{1 + 2} - k P^{2} (- k)) t$

Step 2.2.3.5.1.4.3

Add $1$ and $2$ .

$π = (k P^{2} L - k^{2} P L - k P^{3} - k P^{2} (- k)) t$

Step 2.2.3.5.1.5

Multiply $k$ by $k$ by adding the exponents.

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Step 2.2.3.5.1.5.1

Move $k$ .

$π = (k P^{2} L - k^{2} P L - k P^{3} - (k \cdot k) P^{2} \cdot - 1) t$

Step 2.2.3.5.1.5.2

Multiply $k$ by $k$ .

$π = (k P^{2} L - k^{2} P L - k P^{3} - k^{2} P^{2} \cdot - 1) t$

Step 2.2.3.5.1.6

Multiply $- k^{2} P^{2} \cdot - 1$ .

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Step 2.2.3.5.1.6.1

Multiply $- 1$ by $- 1$ .

$π = (k P^{2} L - k^{2} P L - k P^{3} + 1 k^{2} P^{2}) t$

Step 2.2.3.5.1.6.2

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

$π = (k P^{2} L - k^{2} P L - k P^{3} + k^{2} P^{2}) t$

Step 2.2.3.5.2

Apply the distributive property.

$π = k P^{2} L t - k^{2} P L t - k P^{3} t + k^{2} P^{2} t$

Step 2.3

Solve the equation.

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Step 2.3.1

Rewrite the equation as $k P^{2} L t - k^{2} P L t - k P^{3} t + k^{2} P^{2} t = π$ .

$k P^{2} L t - k^{2} P L t - k P^{3} t + k^{2} P^{2} t = π$

Step 2.3.2

Factor $k P t$ out of $k P^{2} L t - k^{2} P L t - k P^{3} t + k^{2} P^{2} t$ .

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Step 2.3.2.1

Factor $k P t$ out of $k P^{2} L t$ .

$k P t (P L) - k^{2} P L t - k P^{3} t + k^{2} P^{2} t = π$

Step 2.3.2.2

Factor $k P t$ out of $- k^{2} P L t$ .

$k P t (P L) + k P t (- k L) - k P^{3} t + k^{2} P^{2} t = π$

Step 2.3.2.3

Factor $k P t$ out of $- k P^{3} t$ .

$k P t (P L) + k P t (- k L) + k P t (- 1 P^{2}) + k^{2} P^{2} t = π$

Step 2.3.2.4

Factor $k P t$ out of $k^{2} P^{2} t$ .

$k P t (P L) + k P t (- k L) + k P t (- 1 P^{2}) + k P t (k P) = π$

Step 2.3.2.5

Factor $k P t$ out of $k P t (P L) + k P t (- k L)$ .

$k P t (P L - k L) + k P t (- 1 P^{2}) + k P t (k P) = π$

Step 2.3.2.6

Factor $k P t$ out of $k P t (P L - k L) + k P t (- 1 P^{2})$ .

$k P t (P L - k L - 1 P^{2}) + k P t (k P) = π$

Step 2.3.2.7

Factor $k P t$ out of $k P t (P L - k L - 1 P^{2}) + k P t (k P)$ .

$k P t (P L - k L - 1 P^{2} + k P) = π$

Step 2.3.3

Factor out the greatest common factor from each group.

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Step 2.3.3.1

Group the first two terms and the last two terms.

$k P t ((P L - k L) - 1 P^{2} + k P) = π$

Step 2.3.3.2

Factor out the greatest common factor (GCF) from each group.

$k P t (L (P - k) - 1 P (P - k)) = π$

Step 2.3.4

Factor the polynomial by factoring out the greatest common factor, $P - k$ .

$k P t ((P - k) (L - 1 P)) = π$

Step 2.3.5

Factor.

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Step 2.3.5.1

Rewrite $- 1 P$ as $- P$ .

$k P t ((P - k) (L - P)) = π$

Step 2.3.5.2

Remove unnecessary parentheses.

$k P t (P - k) (L - P) = π$

Step 2.3.6

Divide each term in $k P t (P - k) (L - P) = π$ by $k P (P - k) (L - P)$ and simplify.

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Step 2.3.6.1

Divide each term in $k P t (P - k) (L - P) = π$ by $k P (P - k) (L - P)$ .

$\frac{k P t (P - k) (L - P)}{k P (P - k) (L - P)} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2

Simplify the left side.

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Step 2.3.6.2.1

Cancel the common factor of $k$ .

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Step 2.3.6.2.1.1

Cancel the common factor.

$\frac{k P t (P - k) (L - P)}{k P (P - k) (L - P)} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2.1.2

Rewrite the expression.

$\frac{(P t (P - k)) (L - P)}{(P (P - k)) (L - P)} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2.2

Cancel the common factor of $P$ .

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Step 2.3.6.2.2.1

Cancel the common factor.

$\frac{P t (P - k) (L - P)}{P (P - k) (L - P)} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2.2.2

Rewrite the expression.

$\frac{(t (P - k)) (L - P)}{(P - k) (L - P)} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2.3

Cancel the common factor of $P - k$ .

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Step 2.3.6.2.3.1

Cancel the common factor.

$\frac{t (P - k) (L - P)}{(P - k) (L - P)} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2.3.2

Rewrite the expression.

$\frac{(t) (L - P)}{L - P} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2.4

Cancel the common factor of $L - P$ .

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Step 2.3.6.2.4.1

Cancel the common factor.

$\frac{t (L - P)}{L - P} = \frac{π}{k P (P - k) (L - P)}$

Step 2.3.6.2.4.2

Divide $t$ by $1$ .

$t = \frac{π}{k P (P - k) (L - P)}$