Calculus Examples

$y = \frac{42}{\sqrt{49 + x^{2}}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{49 + x^{2}}$ as ${(49 + x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{42}{{(49 + x^{2})}^{\frac{1}{2}}}]$

Step 1.2

Since $42$ is constant with respect to $x$ , the derivative of $\frac{42}{{(49 + x^{2})}^{\frac{1}{2}}}$ with respect to $x$ is $42 \frac{d}{d x} [\frac{1}{{(49 + x^{2})}^{\frac{1}{2}}}]$ .

$42 \frac{d}{d x} [\frac{1}{{(49 + x^{2})}^{\frac{1}{2}}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Rewrite $\frac{1}{{(49 + x^{2})}^{\frac{1}{2}}}$ as ${({(49 + x^{2})}^{\frac{1}{2}})}^{- 1}$ .

$42 \frac{d}{d x} [{({(49 + x^{2})}^{\frac{1}{2}})}^{- 1}]$

Step 1.3.2

Multiply the exponents in ${({(49 + x^{2})}^{\frac{1}{2}})}^{- 1}$ .

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Step 1.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$42 \frac{d}{d x} [{(49 + x^{2})}^{\frac{1}{2} \cdot - 1}]$

Step 1.3.2.2

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

$42 \frac{d}{d x} [{(49 + x^{2})}^{\frac{- 1}{2}}]$

Step 1.3.2.3

Move the negative in front of the fraction.

$42 \frac{d}{d x} [{(49 + x^{2})}^{- \frac{1}{2}}]$

Step 2

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 $g (x) = 49 + x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $49 + x^{2}$ .

$42 (\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d x} [49 + x^{2}])$

Step 2.2

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

$42 (- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d x} [49 + x^{2}])$

Step 2.3

Replace all occurrences of $u$ with $49 + x^{2}$ .

$42 (- \frac{1}{2} {(49 + x^{2})}^{- \frac{1}{2} - 1} \frac{d}{d x} [49 + x^{2}])$

Step 3

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

$42 (- \frac{1}{2} {(49 + x^{2})}^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [49 + x^{2}])$

Step 4

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

$42 (- \frac{1}{2} {(49 + x^{2})}^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [49 + x^{2}])$

Step 5

Combine the numerators over the common denominator.

$42 (- \frac{1}{2} {(49 + x^{2})}^{\frac{- 1 - 1 \cdot 2}{2}} \frac{d}{d x} [49 + x^{2}])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$42 (- \frac{1}{2} {(49 + x^{2})}^{\frac{- 1 - 2}{2}} \frac{d}{d x} [49 + x^{2}])$

Step 6.2

Subtract $2$ from $- 1$ .

$42 (- \frac{1}{2} {(49 + x^{2})}^{\frac{- 3}{2}} \frac{d}{d x} [49 + x^{2}])$

Step 7

Move the negative in front of the fraction.

$42 (- \frac{1}{2} {(49 + x^{2})}^{- \frac{3}{2}} \frac{d}{d x} [49 + x^{2}])$

Step 8

Combine ${(49 + x^{2})}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$42 (- \frac{{(49 + x^{2})}^{- \frac{3}{2}}}{2} \frac{d}{d x} [49 + x^{2}])$

Step 9

Simplify the expression.

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Step 9.1

Move ${(49 + x^{2})}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$42 (- \frac{1}{2 {(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [49 + x^{2}])$

Step 9.2

Multiply $- 1$ by $42$ .

$- 42 (\frac{1}{2 {(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [49 + x^{2}])$

Step 10

Combine $\frac{1}{2 {(49 + x^{2})}^{\frac{3}{2}}}$ and $- 42$ .

$\frac{- 42}{2 {(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [49 + x^{2}]$

Step 11

Factor $2$ out of $- 42$ .

$\frac{2 \cdot - 21}{2 {(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [49 + x^{2}]$

Step 12

Cancel the common factors.

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Step 12.1

Factor $2$ out of $2 {(49 + x^{2})}^{\frac{3}{2}}$ .

$\frac{2 \cdot - 21}{2 ({(49 + x^{2})}^{\frac{3}{2}})} \frac{d}{d x} [49 + x^{2}]$

Step 12.2

Cancel the common factor.

$\frac{2 \cdot - 21}{2 {(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [49 + x^{2}]$

Step 12.3

Rewrite the expression.

$\frac{- 21}{{(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [49 + x^{2}]$

Step 13

Move the negative in front of the fraction.

$- \frac{21}{{(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [49 + x^{2}]$

Step 14

By the Sum Rule, the derivative of $49 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [49] + \frac{d}{d x} [x^{2}]$ .

$- \frac{21}{{(49 + x^{2})}^{\frac{3}{2}}} (\frac{d}{d x} [49] + \frac{d}{d x} [x^{2}])$

Step 15

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$- \frac{21}{{(49 + x^{2})}^{\frac{3}{2}}} (0 + \frac{d}{d x} [x^{2}])$

Step 16

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

$- \frac{21}{{(49 + x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [x^{2}]$

Step 17

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

$- \frac{21}{{(49 + x^{2})}^{\frac{3}{2}}} (2 x)$

Step 18

Combine fractions.

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Step 18.1

Multiply $2$ by $- 1$ .

$- 2 \frac{21}{{(49 + x^{2})}^{\frac{3}{2}}} x$

Step 18.2

Combine $- 2$ and $\frac{21}{{(49 + x^{2})}^{\frac{3}{2}}}$ .

$\frac{- 2 \cdot 21}{{(49 + x^{2})}^{\frac{3}{2}}} x$

Step 18.3

Multiply $- 2$ by $21$ .

$\frac{- 42}{{(49 + x^{2})}^{\frac{3}{2}}} x$

Step 18.4

Combine $\frac{- 42}{{(49 + x^{2})}^{\frac{3}{2}}}$ and $x$ .

$\frac{- 42 x}{{(49 + x^{2})}^{\frac{3}{2}}}$

Step 18.5

Move the negative in front of the fraction.

$- \frac{42 x}{{(49 + x^{2})}^{\frac{3}{2}}}$