Calculus Examples

$\frac{d}{d x} {(4 + (f (8 x)))}^{3}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 4 + f (8 x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 + f (8 x)$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [4 + f (8 x)]$

Step 1.2

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

$3 {(4 + f (8 x))}^{2} \frac{d}{d x} [4 + f (8 x)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 + f (8 x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [f (8 x)]$ .

$3 {(4 + f (8 x))}^{2} (\frac{d}{d x} [4] + \frac{d}{d x} [f (8 x)])$

Step 2.2

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

$3 {(4 + f (8 x))}^{2} (0 + \frac{d}{d x} [f (8 x)])$

Step 2.3

Add $0$ and $\frac{d}{d x} [f (8 x)]$ .

$3 {(4 + f (8 x))}^{2} \frac{d}{d x} [f (8 x)]$