Calculus Examples

$\sin^{2} (6 x)$

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 $g (x) = \sin (6 x)$ .

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To apply the Chain Rule, set $u_{1}$ as $\sin (6 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (6 x)]$

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

$2 u_{1} \frac{d}{d x} [\sin (6 x)]$

Replace all occurrences of $u_{1}$ with $\sin (6 x)$ .

$2 \sin (6 x) \frac{d}{d x} [\sin (6 x)]$

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 $g (x) = 6 x$ .

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To apply the Chain Rule, set $u_{2}$ as $6 x$ .

$2 \sin (6 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [6 x])$

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

$2 \sin (6 x) (\cos (u_{2}) \frac{d}{d x} [6 x])$

Replace all occurrences of $u_{2}$ with $6 x$ .

$2 \sin (6 x) (\cos (6 x) \frac{d}{d x} [6 x])$

Differentiate.

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Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$2 \sin (6 x) (\cos (6 x) (6 \frac{d}{d x} [x]))$

Move $6$ to the left of $\cos (6 x)$ .

$2 \sin (6 x) (6 \cdot \cos (6 x) \frac{d}{d x} [x])$

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

$2 \sin (6 x) (6 \cos (6 x) \cdot 1)$

Multiply $6$ by $1$ .

$2 \sin (6 x) (6 \cos (6 x))$

Simplify.

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Multiply $6$ by $2$ .

$12 \sin (6 x) \cos (6 x)$

Reorder the factors of $12 \sin (6 x) \cos (6 x)$ .

$12 \cos (6 x) \sin (6 x)$

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