Calculus Examples

$y = 3 {(2 - x^{2})}^{4}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 {(2 - x^{2})}^{4}$ with respect to $x$ is $3 \frac{d}{d x} [{(2 - x^{2})}^{4}]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 2 - x^{2}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Multiply $4$ by $3$ .

$12 ({(2 - x^{2})}^{3} \frac{d}{d x} [2 - x^{2}])$

Step 3.2

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

$12 {(2 - x^{2})}^{3} (\frac{d}{d x} [2] + \frac{d}{d x} [- x^{2}])$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$12 {(2 - x^{2})}^{3} (0 + \frac{d}{d x} [- x^{2}])$

Step 3.4

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

$12 {(2 - x^{2})}^{3} \frac{d}{d x} [- x^{2}]$

Step 3.5

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

$12 {(2 - x^{2})}^{3} (- \frac{d}{d x} [x^{2}])$

Step 3.6

Multiply $- 1$ by $12$ .

$- 12 {(2 - x^{2})}^{3} \frac{d}{d x} [x^{2}]$

Step 3.7

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

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

Step 3.8

Simplify the expression.

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Step 3.8.1

Multiply $2$ by $- 12$ .

$- 24 {(2 - x^{2})}^{3} x$

Step 3.8.2

Reorder the factors of $- 24 {(2 - x^{2})}^{3} x$ .

$- 24 x {(2 - x^{2})}^{3}$