Calculus Examples

$y = \sqrt{\frac{x^{2} - 5}{x^{2} + 4}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{2} - 5}{x^{2} + 4}}$ as ${(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1}{2}}$ .

$y = {(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \frac{x^{2} - 5}{x^{2} + 4}$ .

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Step 4.1.1

To apply the Chain Rule, set $u$ as $\frac{x^{2} - 5}{x^{2} + 4}$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.1.3

Replace all occurrences of $u$ with $\frac{x^{2} - 5}{x^{2} + 4}$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{1 - 2}{2}} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{\frac{- 1}{2}} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.6

Move the negative in front of the fraction.

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{x^{2} - 5}{x^{2} + 4}]$

Step 4.7

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 $f (x) = x^{2} - 5$ and $g (x) = x^{2} + 4$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{(x^{2} + 4) \frac{d}{d x} [x^{2} - 5] - (x^{2} - 5) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.8

Differentiate.

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Step 4.8.1

By the Sum Rule, the derivative of $x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{(x^{2} + 4) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]) - (x^{2} - 5) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.8.2

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

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{(x^{2} + 4) (2 x + \frac{d}{d x} [- 5]) - (x^{2} - 5) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.8.3

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{(x^{2} + 4) (2 x + 0) - (x^{2} - 5) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.8.4

Simplify the expression.

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Step 4.8.4.1

Add $2 x$ and $0$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{(x^{2} + 4) (2 x) - (x^{2} - 5) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.8.4.2

Move $2$ to the left of $x^{2} + 4$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{2 \cdot (x^{2} + 4) x - (x^{2} - 5) \frac{d}{d x} [x^{2} + 4]}{{(x^{2} + 4)}^{2}}$

Step 4.8.5

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{2 (x^{2} + 4) x - (x^{2} - 5) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])}{{(x^{2} + 4)}^{2}}$

Step 4.8.6

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

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{2 (x^{2} + 4) x - (x^{2} - 5) (2 x + \frac{d}{d x} [4])}{{(x^{2} + 4)}^{2}}$

Step 4.8.7

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{2 (x^{2} + 4) x - (x^{2} - 5) (2 x + 0)}{{(x^{2} + 4)}^{2}}$

Step 4.8.8

Combine fractions.

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Step 4.8.8.1

Add $2 x$ and $0$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{2 (x^{2} + 4) x - (x^{2} - 5) (2 x)}{{(x^{2} + 4)}^{2}}$

Step 4.8.8.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}} \frac{2 (x^{2} + 4) x - 2 (x^{2} - 5) x}{{(x^{2} + 4)}^{2}}$

Step 4.8.8.3

Multiply $\frac{2 (x^{2} + 4) x - 2 (x^{2} - 5) x}{{(x^{2} + 4)}^{2}}$ by $\frac{1}{2}$ .

$\frac{2 (x^{2} + 4) x - 2 (x^{2} - 5) x}{{(x^{2} + 4)}^{2} \cdot 2} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}}$

Step 4.8.8.4

Move $2$ to the left of ${(x^{2} + 4)}^{2}$ .

$\frac{2 (x^{2} + 4) x - 2 (x^{2} - 5) x}{2 \cdot {(x^{2} + 4)}^{2}} {(\frac{x^{2} - 5}{x^{2} + 4})}^{- \frac{1}{2}}$

Step 4.9

Simplify.

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Step 4.9.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{2 (x^{2} + 4) x - 2 (x^{2} - 5) x}{2 {(x^{2} + 4)}^{2}} {(\frac{x^{2} + 4}{x^{2} - 5})}^{\frac{1}{2}}$

Step 4.9.2

Apply the product rule to $\frac{x^{2} + 4}{x^{2} - 5}$ .

$\frac{2 (x^{2} + 4) x - 2 (x^{2} - 5) x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.3

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot 4) x - 2 (x^{2} - 5) x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.4

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 4 x - 2 (x^{2} - 5) x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.5

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 4 x + (- 2 x^{2} - 2 \cdot - 5) x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.6

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 4 x - 2 x^{2} x - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7

Combine terms.

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Step 4.9.7.1

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot 4 x - 2 x^{2} x - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.2

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

$\frac{2 x^{1 + 2} + 2 \cdot 4 x - 2 x^{2} x - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot 4 x - 2 x^{2} x - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.4

Multiply $2$ by $4$ .

$\frac{2 x^{3} + 8 x - 2 x^{2} x - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.5

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

$\frac{2 x^{3} + 8 x - 2 (x^{1} x^{2}) - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.6

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

$\frac{2 x^{3} + 8 x - 2 x^{1 + 2} - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.7

Add $1$ and $2$ .

$\frac{2 x^{3} + 8 x - 2 x^{3} - 2 \cdot - 5 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.8

Multiply $- 2$ by $- 5$ .

$\frac{2 x^{3} + 8 x - 2 x^{3} + 10 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.9

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

$\frac{8 x + 0 + 10 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.10

Add $8 x$ and $0$ .

$\frac{8 x + 10 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.11

Add $8 x$ and $10 x$ .

$\frac{18 x}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.12

Cancel the common factor of $18$ and $2$ .

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Step 4.9.7.12.1

Factor $2$ out of $18 x$ .

$\frac{2 (9 x)}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.12.2

Cancel the common factors.

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Step 4.9.7.12.2.1

Factor $2$ out of $2 {(x^{2} + 4)}^{2}$ .

$\frac{2 (9 x)}{2 ({(x^{2} + 4)}^{2})} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.12.2.2

Cancel the common factor.

$\frac{2 (9 x)}{2 {(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.12.2.3

Rewrite the expression.

$\frac{9 x}{{(x^{2} + 4)}^{2}} \cdot \frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.13

Multiply $\frac{9 x}{{(x^{2} + 4)}^{2}}$ by $\frac{{(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} - 5)}^{\frac{1}{2}}}$ .

$\frac{9 x {(x^{2} + 4)}^{\frac{1}{2}}}{{(x^{2} + 4)}^{2} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.14

Move ${(x^{2} + 4)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{9 x}{{(x^{2} + 4)}^{2} {(x^{2} - 5)}^{\frac{1}{2}} {(x^{2} + 4)}^{- \frac{1}{2}}}$

Step 4.9.7.15

Multiply ${(x^{2} + 4)}^{2}$ by ${(x^{2} + 4)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.9.7.15.1

Move ${(x^{2} + 4)}^{- \frac{1}{2}}$ .

$\frac{9 x}{{(x^{2} + 4)}^{- \frac{1}{2}} {(x^{2} + 4)}^{2} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.15.2

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

$\frac{9 x}{{(x^{2} + 4)}^{- \frac{1}{2} + 2} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.15.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{9 x}{{(x^{2} + 4)}^{- \frac{1}{2} + 2 \cdot \frac{2}{2}} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.15.4

Combine $2$ and $\frac{2}{2}$ .

$\frac{9 x}{{(x^{2} + 4)}^{- \frac{1}{2} + \frac{2 \cdot 2}{2}} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.15.5

Combine the numerators over the common denominator.

$\frac{9 x}{{(x^{2} + 4)}^{\frac{- 1 + 2 \cdot 2}{2}} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.15.6

Simplify the numerator.

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Step 4.9.7.15.6.1

Multiply $2$ by $2$ .

$\frac{9 x}{{(x^{2} + 4)}^{\frac{- 1 + 4}{2}} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 4.9.7.15.6.2

Add $- 1$ and $4$ .

$\frac{9 x}{{(x^{2} + 4)}^{\frac{3}{2}} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 5

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

$y' = \frac{9 x}{{(x^{2} + 4)}^{\frac{3}{2}} {(x^{2} - 5)}^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{9 x}{{(x^{2} + 4)}^{\frac{3}{2}} {(x^{2} - 5)}^{\frac{1}{2}}}$