Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{3} - 2 x$ .

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

Step 1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

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

Step 1.3

Replace all occurrences of $u$ with $x^{3} - 2 x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} - 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2 x]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$- \sin (x^{3} - 2 x) (3 x^{2} - 2 \cdot 1)$

Step 2.5

Multiply $- 2$ by $1$ .

$- \sin (x^{3} - 2 x) (3 x^{2} - 2)$