Calculus Examples

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

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 $g (x) = \sin (3 x - 2)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (3 x - 2)$ .

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

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 $n = 2$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (3 x - 2)$ .

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

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 $g (x) = 3 x - 2$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$2 \sin (3 x - 2) \cos (3 x - 2) (3 \cdot 1 + \frac{d}{d x} [- 2])$

Step 3.4

Multiply $3$ by $1$ .

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

Step 3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 \sin (3 x - 2) \cos (3 x - 2) (3 + 0)$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

$2 \sin (3 x - 2) \cos (3 x - 2) \cdot 3$

Step 3.6.2

Multiply $3$ by $2$ .

$6 \sin (3 x - 2) \cos (3 x - 2)$

Step 3.6.3

Reorder the factors of $6 \sin (3 x - 2) \cos (3 x - 2)$ .

$6 \cos (3 x - 2) \sin (3 x - 2)$