Calculus Examples

$\frac{1}{{(25 - x^{2})}^{3}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\frac{1}{{(25 - x^{2})}^{3}}$ as ${({(25 - x^{2})}^{3})}^{- 1}$ .

$\frac{d}{d x} [{({(25 - x^{2})}^{3})}^{- 1}]$

Step 1.2

Multiply the exponents in ${({(25 - x^{2})}^{3})}^{- 1}$ .

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Step 1.2.1

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

$\frac{d}{d x} [{(25 - x^{2})}^{3 \cdot - 1}]$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\frac{d}{d x} [{(25 - x^{2})}^{- 3}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = 25 - x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $25 - x^{2}$ .

$\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [25 - x^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 3$ .

$- 3 u^{- 4} \frac{d}{d x} [25 - x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $25 - x^{2}$ .

$- 3 {(25 - x^{2})}^{- 4} \frac{d}{d x} [25 - x^{2}]$

Step 3

Differentiate.

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Step 3.1

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

$- 3 {(25 - x^{2})}^{- 4} (\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}])$

Step 3.2

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

$- 3 {(25 - x^{2})}^{- 4} (0 + \frac{d}{d x} [- x^{2}])$

Step 3.3

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

$- 3 {(25 - x^{2})}^{- 4} \frac{d}{d x} [- x^{2}]$

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$- 3 {(25 - x^{2})}^{- 4} (- \frac{d}{d x} [x^{2}])$

Step 3.5

Multiply $- 1$ by $- 3$ .

$3 {(25 - x^{2})}^{- 4} \frac{d}{d x} [x^{2}]$

Step 3.6

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

$3 {(25 - x^{2})}^{- 4} (2 x)$

Step 3.7

Multiply $2$ by $3$ .

$6 {(25 - x^{2})}^{- 4} x$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 \frac{1}{{(25 - x^{2})}^{4}} x$

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

Combine $6$ and $\frac{1}{{(25 - x^{2})}^{4}}$ .

$\frac{6}{{(25 - x^{2})}^{4}} x$

Step 5.1.2

Combine $\frac{6}{{(25 - x^{2})}^{4}}$ and $x$ .

$\frac{6 x}{{(25 - x^{2})}^{4}}$

Step 5.2

Reorder terms.

$\frac{6 x}{{(- x^{2} + 25)}^{4}}$