Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (3 x)$ .

$2 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (3 x)])$

Step 2.2

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

$2 (2 u_{1} \frac{d}{d x} [\sin (3 x)])$

Step 2.3

Replace all occurrences of $u_{1}$ with $\sin (3 x)$ .

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

Step 3

Multiply $2$ by $2$ .

$4 (\sin (3 x) \frac{d}{d x} [\sin (3 x)])$

Step 4

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 $g (x) = 3 x$ .

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Step 4.1

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

$4 \sin (3 x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [3 x])$

Step 4.2

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

$4 \sin (3 x) (\cos (u_{2}) \frac{d}{d x} [3 x])$

Step 4.3

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

$4 \sin (3 x) (\cos (3 x) \frac{d}{d x} [3 x])$

Step 5

Differentiate.

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Step 5.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$4 \sin (3 x) \cos (3 x) (3 \frac{d}{d x} [x])$

Step 5.2

Multiply $3$ by $4$ .

$12 \sin (3 x) \cos (3 x) \frac{d}{d x} [x]$

Step 5.3

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

$12 \sin (3 x) \cos (3 x) \cdot 1$

Step 5.4

Simplify the expression.

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Step 5.4.1

Multiply $12$ by $1$ .

$12 \sin (3 x) \cos (3 x)$

Step 5.4.2

Reorder the factors of $12 \sin (3 x) \cos (3 x)$ .

$12 \cos (3 x) \sin (3 x)$