Calculus Examples

${(5 - 8 x^{2})}^{7}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = 5 - 8 x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $5 - 8 x^{2}$ .

$\frac{d}{d u} [u^{7}] \frac{d}{d x} [5 - 8 x^{2}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$7 u^{6} \frac{d}{d x} [5 - 8 x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $5 - 8 x^{2}$ .

$7 {(5 - 8 x^{2})}^{6} \frac{d}{d x} [5 - 8 x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 - 8 x^{2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 8 x^{2}]$ .

$7 {(5 - 8 x^{2})}^{6} (\frac{d}{d x} [5] + \frac{d}{d x} [- 8 x^{2}])$

Step 2.2

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

$7 {(5 - 8 x^{2})}^{6} (0 + \frac{d}{d x} [- 8 x^{2}])$

Step 2.3

Add $0$ and $\frac{d}{d x} [- 8 x^{2}]$ .

$7 {(5 - 8 x^{2})}^{6} \frac{d}{d x} [- 8 x^{2}]$

Step 2.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{2}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{2}]$ .

$7 {(5 - 8 x^{2})}^{6} (- 8 \frac{d}{d x} [x^{2}])$

Step 2.5

Multiply $- 8$ by $7$ .

$- 56 {(5 - 8 x^{2})}^{6} \frac{d}{d x} [x^{2}]$

Step 2.6

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

$- 56 {(5 - 8 x^{2})}^{6} (2 x)$

Step 2.7

Simplify the expression.

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Step 2.7.1

Multiply $2$ by $- 56$ .

$- 112 {(5 - 8 x^{2})}^{6} x$

Step 2.7.2

Reorder the factors of $- 112 {(5 - 8 x^{2})}^{6} x$ .

$- 112 x {(5 - 8 x^{2})}^{6}$