Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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 $g (x) = x^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

Step 2.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

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

Step 2.2.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $3$ .

$6 \cos (x^{2}) x + \frac{d}{d x} [- 2 \cos (3 x)]$

Step 3

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

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Step 3.1

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

$6 \cos (x^{2}) x - 2 \frac{d}{d x} [\cos (3 x)]$

Step 3.2

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

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Step 3.2.1

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

$6 \cos (x^{2}) x - 2 (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [3 x])$

Step 3.2.2

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

$6 \cos (x^{2}) x - 2 (- \sin (u_{2}) \frac{d}{d x} [3 x])$

Step 3.2.3

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

$6 \cos (x^{2}) x - 2 (- \sin (3 x) \frac{d}{d x} [3 x])$

Step 3.3

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

$6 \cos (x^{2}) x - 2 (- \sin (3 x) (3 \frac{d}{d x} [x]))$

Step 3.4

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

$6 \cos (x^{2}) x - 2 (- \sin (3 x) (3 \cdot 1))$

Step 3.5

Multiply $3$ by $1$ .

$6 \cos (x^{2}) x - 2 (- \sin (3 x) \cdot 3)$

Step 3.6

Multiply $3$ by $- 1$ .

$6 \cos (x^{2}) x - 2 (- 3 \sin (3 x))$

Step 3.7

Multiply $- 3$ by $- 2$ .

$6 \cos (x^{2}) x + 6 \sin (3 x)$

Step 4

Reorder terms.

$6 x \cos (x^{2}) + 6 \sin (3 x)$