Calculus Examples

$\frac{x^{2} + 2 x - 35}{x^{2} - 25}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} + 2 x - 35$ and $g (x) = x^{2} - 25$ .

$\frac{(x^{2} - 25) \frac{d}{d x} [x^{2} + 2 x - 35] - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} + 2 x - 35$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 35]$ .

$\frac{(x^{2} - 25) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 35]) - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2.2

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

$\frac{(x^{2} - 25) (2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 35]) - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{(x^{2} - 25) (2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 35]) - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2.4

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

$\frac{(x^{2} - 25) (2 x + 2 \cdot 1 + \frac{d}{d x} [- 35]) - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2.5

Multiply $2$ by $1$ .

$\frac{(x^{2} - 25) (2 x + 2 + \frac{d}{d x} [- 35]) - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2.6

Since $- 35$ is constant with respect to $x$ , the derivative of $- 35$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 25) (2 x + 2 + 0) - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2.7

Add $2 x + 2$ and $0$ .

$\frac{(x^{2} - 25) (2 x + 2) - (x^{2} + 2 x - 35) \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 2.8

By the Sum Rule, the derivative of $x^{2} - 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25]$ .

$\frac{(x^{2} - 25) (2 x + 2) - (x^{2} + 2 x - 35) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25])}{{(x^{2} - 25)}^{2}}$

Step 2.9

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

$\frac{(x^{2} - 25) (2 x + 2) - (x^{2} + 2 x - 35) (2 x + \frac{d}{d x} [- 25])}{{(x^{2} - 25)}^{2}}$

Step 2.10

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 25) (2 x + 2) - (x^{2} + 2 x - 35) (2 x + 0)}{{(x^{2} - 25)}^{2}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $2 x$ and $0$ .

$\frac{(x^{2} - 25) (2 x + 2) - (x^{2} + 2 x - 35) (2 x)}{{(x^{2} - 25)}^{2}}$

Step 2.11.2

Multiply $2$ by $- 1$ .

$\frac{(x^{2} - 25) (2 x + 2) - 2 (x^{2} + 2 x - 35) x}{{(x^{2} - 25)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x^{2} - 25) (2 x + 2) + (- 2 x^{2} - 2 (2 x) - 2 \cdot - 35) x}{{(x^{2} - 25)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{(x^{2} - 25) (2 x + 2) - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Expand $(x^{2} - 25) (2 x + 2)$ using the FOIL Method.

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Step 3.3.1.1.1

Apply the distributive property.

$\frac{x^{2} (2 x + 2) - 25 (2 x + 2) - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.1.2

Apply the distributive property.

$\frac{x^{2} (2 x) + x^{2} \cdot 2 - 25 (2 x + 2) - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.1.3

Apply the distributive property.

$\frac{x^{2} (2 x) + x^{2} \cdot 2 - 25 (2 x) - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2

Simplify each term.

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Step 3.3.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} x + x^{2} \cdot 2 - 25 (2 x) - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2.2

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

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Step 3.3.1.2.2.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + x^{2} \cdot 2 - 25 (2 x) - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2.2.2

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

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Step 3.3.1.2.2.2.1

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

$\frac{2 (x^{1} x^{2}) + x^{2} \cdot 2 - 25 (2 x) - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2.2.2.2

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

$\frac{2 x^{1 + 2} + x^{2} \cdot 2 - 25 (2 x) - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot 2 - 25 (2 x) - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2.3

Move $2$ to the left of $x^{2}$ .

$\frac{2 x^{3} + 2 \cdot x^{2} - 25 (2 x) - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2.4

Multiply $2$ by $- 25$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 25 \cdot 2 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.2.5

Multiply $- 25$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{2} x - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.3

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

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Step 3.3.1.3.1

Move $x$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 (x \cdot x^{2}) - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.3.2

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

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Step 3.3.1.3.2.1

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

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 (x^{1} x^{2}) - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.3.2.2

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

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{1 + 2} - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.3.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{3} - 2 (2 x) x - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.4.1

Move $x$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{3} - 2 (2 (x \cdot x)) - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.4.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{3} - 2 (2 x^{2}) - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.5

Multiply $2$ by $- 2$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{3} - 4 x^{2} - 2 \cdot - 35 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.1.6

Multiply $- 2$ by $- 35$ .

$\frac{2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{3} - 4 x^{2} + 70 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.2

Combine the opposite terms in $2 x^{3} + 2 x^{2} - 50 x - 50 - 2 x^{3} - 4 x^{2} + 70 x$ .

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Step 3.3.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{2 x^{2} - 50 x - 50 + 0 - 4 x^{2} + 70 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.2.2

Add $2 x^{2} - 50 x - 50$ and $0$ .

$\frac{2 x^{2} - 50 x - 50 - 4 x^{2} + 70 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.3

Subtract $4 x^{2}$ from $2 x^{2}$ .

$\frac{- 2 x^{2} - 50 x - 50 + 70 x}{{(x^{2} - 25)}^{2}}$

Step 3.3.4

Add $- 50 x$ and $70 x$ .

$\frac{- 2 x^{2} + 20 x - 50}{{(x^{2} - 25)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $2$ out of $- 2 x^{2} + 20 x - 50$ .

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Step 3.4.1.1

Factor $2$ out of $- 2 x^{2}$ .

$\frac{2 (- x^{2}) + 20 x - 50}{{(x^{2} - 25)}^{2}}$

Step 3.4.1.2

Factor $2$ out of $20 x$ .

$\frac{2 (- x^{2}) + 2 (10 x) - 50}{{(x^{2} - 25)}^{2}}$

Step 3.4.1.3

Factor $2$ out of $- 50$ .

$\frac{2 (- x^{2}) + 2 (10 x) + 2 (- 25)}{{(x^{2} - 25)}^{2}}$

Step 3.4.1.4

Factor $2$ out of $2 (- x^{2}) + 2 (10 x)$ .

$\frac{2 (- x^{2} + 10 x) + 2 (- 25)}{{(x^{2} - 25)}^{2}}$

Step 3.4.1.5

Factor $2$ out of $2 (- x^{2} + 10 x) + 2 (- 25)$ .

$\frac{2 (- x^{2} + 10 x - 25)}{{(x^{2} - 25)}^{2}}$

Step 3.4.2

Factor by grouping.

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Step 3.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 25 = 25$ and whose sum is $b = 10$ .

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Step 3.4.2.1.1

Factor $10$ out of $10 x$ .

$\frac{2 (- x^{2} + 10 (x) - 25)}{{(x^{2} - 25)}^{2}}$

Step 3.4.2.1.2

Rewrite $10$ as $5$ plus $5$

$\frac{2 (- x^{2} + (5 + 5) x - 25)}{{(x^{2} - 25)}^{2}}$

Step 3.4.2.1.3

Apply the distributive property.

$\frac{2 (- x^{2} + 5 x + 5 x - 25)}{{(x^{2} - 25)}^{2}}$

Step 3.4.2.2

Factor out the greatest common factor from each group.

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Step 3.4.2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((- x^{2} + 5 x) + 5 x - 25)}{{(x^{2} - 25)}^{2}}$

Step 3.4.2.2.2

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

$\frac{2 (x (- x + 5) - 5 (- x + 5))}{{(x^{2} - 25)}^{2}}$

Step 3.4.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 5$ .

$\frac{2 ((- x + 5) (x - 5))}{{(x^{2} - 25)}^{2}}$

$\frac{2 (- x + 5) (x - 5)}{{(x^{2} - 25)}^{2}}$

Step 3.4.3

Combine exponents.

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Step 3.4.3.1

Factor $- 1$ out of $- x$ .

$\frac{2 (- (x) + 5) (x - 5)}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.2

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{2 (- (x) - 1 (- 5)) (x - 5)}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.3

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{2 (- (x - 5)) (x - 5)}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.4

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{2 (- 1 (x - 5)) (x - 5)}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.5

Raise $x - 5$ to the power of $1$ .

$\frac{2 \cdot - 1 ({(x - 5)}^{1} (x - 5))}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.6

Raise $x - 5$ to the power of $1$ .

$\frac{2 \cdot - 1 ({(x - 5)}^{1} {(x - 5)}^{1})}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.7

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

$\frac{2 \cdot - 1 {(x - 5)}^{1 + 1}}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.8

Add $1$ and $1$ .

$\frac{2 \cdot - 1 {(x - 5)}^{2}}{{(x^{2} - 25)}^{2}}$

Step 3.4.3.9

Multiply $2$ by $- 1$ .

$\frac{- 2 {(x - 5)}^{2}}{{(x^{2} - 25)}^{2}}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Rewrite $25$ as $5^{2}$ .

$\frac{- 2 {(x - 5)}^{2}}{{(x^{2} - 5^{2})}^{2}}$

Step 3.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{- 2 {(x - 5)}^{2}}{{((x + 5) (x - 5))}^{2}}$

Step 3.5.3

Apply the product rule to $(x + 5) (x - 5)$ .

$\frac{- 2 {(x - 5)}^{2}}{{(x + 5)}^{2} {(x - 5)}^{2}}$

Step 3.6

Cancel the common factor of ${(x - 5)}^{2}$ .

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Step 3.6.1

Cancel the common factor.

$\frac{- 2 {(x - 5)}^{2}}{{(x + 5)}^{2} {(x - 5)}^{2}}$

Step 3.6.2

Rewrite the expression.

$\frac{- 2}{{(x + 5)}^{2}}$

Step 3.7

Move the negative in front of the fraction.

$- \frac{2}{{(x + 5)}^{2}}$