Calculus Examples

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

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 $f (x) = \cos (2 x)$ and $g (x) = \sin (2 x)$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 4

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6

Differentiate.

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Step 6.1

Add $1$ and $1$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Multiply $2$ by $1$ .

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

Step 6.4.2

Move $2$ to the left of $\cos^{2} (2 x)$ .

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

Step 7

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 $g (x) = 2 x$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 9

Raise $\sin (2 x)$ to the power of $1$ .

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

Step 10

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11

Add $1$ and $1$ .

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

Step 12

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

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

Step 13

Multiply $2$ by $- 1$ .

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

Step 14

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

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

Step 15

Multiply $- 2$ by $1$ .

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