Calculus Examples

$f (x) = {(x - 3)}^{3} \tan (x)$

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 $f (x) = {(x - 3)}^{3}$ and $g (x) = \tan (x)$ .

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

Step 2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 3

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 $g (x) = x - 3$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x - 3$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

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

Step 3.3

Replace all occurrences of $u$ with $x - 3$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

${(x - 3)}^{3} \sec^{2} (x) + \tan (x) (3 {(x - 3)}^{2} (1 + 0))$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

${(x - 3)}^{3} \sec^{2} (x) + \tan (x) (3 {(x - 3)}^{2} \cdot 1)$

Step 4.4.2

Multiply $3$ by $1$ .

${(x - 3)}^{3} \sec^{2} (x) + \tan (x) (3 {(x - 3)}^{2})$

Step 4.4.3

Reorder terms.

$\sec^{2} (x) {(x - 3)}^{3} + 3 {(x - 3)}^{2} \tan (x)$