Calculus Examples

$f (x) = \sin^{2} (\cos (2 x))$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (\cos (2 x))$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (\cos (2 x))]$

Step 1.2

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

$2 u_{1} \frac{d}{d x} [\sin (\cos (2 x))]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (\cos (2 x))$ .

$2 \sin (\cos (2 x)) \frac{d}{d x} [\sin (\cos (2 x))]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\cos (2 x)$ .

$2 \sin (\cos (2 x)) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [\cos (2 x)])$

Step 2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$2 \sin (\cos (2 x)) (\cos (u_{2}) \frac{d}{d x} [\cos (2 x)])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\cos (2 x)$ .

$2 \sin (\cos (2 x)) (\cos (\cos (2 x)) \frac{d}{d x} [\cos (2 x)])$

Step 3

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 $g (x) = 2 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

$2 \sin (\cos (2 x)) \cos (\cos (2 x)) (\frac{d}{d u_{3}} [\cos (u_{3})] \frac{d}{d x} [2 x])$

Step 3.2

The derivative of $\cos (u_{3})$ with respect to $u_{3}$ is $- \sin (u_{3})$ .

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

Step 3.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $2$ .

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

Step 4.2

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

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

Step 4.3

Multiply $2$ by $- 2$ .

$- 4 \sin (\cos (2 x)) \cos (\cos (2 x)) \sin (2 x) \frac{d}{d x} [x]$

Step 4.4

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

$- 4 \sin (\cos (2 x)) \cos (\cos (2 x)) \sin (2 x) \cdot 1$

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $- 4$ by $1$ .

$- 4 \sin (\cos (2 x)) \cos (\cos (2 x)) \sin (2 x)$

Step 4.5.2

Reorder the factors of $- 4 \sin (\cos (2 x)) \cos (\cos (2 x)) \sin (2 x)$ .

$- 4 \cos (\cos (2 x)) \sin (\cos (2 x)) \sin (2 x)$