Calculus Examples

$y = \frac{1}{3} \cos^{3} (2 x)$

Step 1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} \cos^{3} (2 x)$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\cos^{3} (2 x)]$ .

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

Step 2

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

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Step 2.1

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

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

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 $n = 3$ .

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

Step 2.3

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

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

Step 3

Simplify terms.

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Step 3.1

Combine $3$ and $\frac{1}{3}$ .

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

Step 3.2

Combine $\cos^{2} (2 x)$ and $\frac{3}{3}$ .

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Divide $\cos^{2} (2 x)$ by $1$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

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 $2 \frac{d}{d x} [x]$ .

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

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

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

Step 5.4

Simplify the expression.

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Step 5.4.1

Multiply $- 2$ by $1$ .

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

Step 5.4.2

Reorder the factors of $\cos^{2} (2 x) (- 2 \sin (2 x))$ .

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