Calculus Examples

$h (x) = {(x^{4} - 1)}^{3}$

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 $g (x) = x^{4} - 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{4} - 1$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [x^{4} - 1]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [x^{4} - 1]$

Step 1.3

Replace all occurrences of $u$ with $x^{4} - 1$ .

$3 {(x^{4} - 1)}^{2} \frac{d}{d x} [x^{4} - 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{4} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 1]$ .

$3 {(x^{4} - 1)}^{2} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 1])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$3 {(x^{4} - 1)}^{2} (4 x^{3} + \frac{d}{d x} [- 1])$

Step 2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$3 {(x^{4} - 1)}^{2} (4 x^{3} + 0)$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $4 x^{3}$ and $0$ .

$3 {(x^{4} - 1)}^{2} (4 x^{3})$

Step 2.4.2

Multiply $4$ by $3$ .

$12 {(x^{4} - 1)}^{2} x^{3}$

Step 2.4.3

Reorder the factors of $12 {(x^{4} - 1)}^{2} x^{3}$ .

$12 x^{3} {(x^{4} - 1)}^{2}$