Calculus Examples

$\cos (x \sin (x))$

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 $g (x) = x \sin (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x \sin (x)$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [x \sin (x)]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x \sin (x)$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \sin (x)$ .

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

Step 3

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

$- \sin (x \sin (x)) (x \cos (x) + \sin (x) \frac{d}{d x} [x])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

$- \sin (x \sin (x)) (x \cos (x) + \sin (x) \cdot 1)$

Step 4.2

Multiply $\sin (x)$ by $1$ .

$- \sin (x \sin (x)) (x \cos (x) + \sin (x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- \sin (x \sin (x)) (x \cos (x)) - \sin (x \sin (x)) \sin (x)$

Step 5.2

Reorder terms.

$- x \cos (x) \sin (x \sin (x)) - \sin (x) \sin (x \sin (x))$