Calculus Examples

$f (x) = \sin (2 x) \cos (3 x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (2 x)$ and $g (x) = \cos (3 x)$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{1}$ with $3 x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Multiply $3$ by $- 1$ .

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

Step 3.3

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

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

Step 3.4

Multiply $- 3$ by $1$ .

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

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 $g (x) = 2 x$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

$\sin (2 x) (- 3 \sin (3 x)) + \cos (3 x) \cos (2 x) (2 \cdot 1)$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $2$ by $1$ .

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

Step 5.3.2

Move $2$ to the left of $\cos (3 x) \cos (2 x)$ .

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

Step 5.3.3

Reorder terms.

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