Calculus Examples

$y = \tan (x) + \frac{1}{3} \tan^{3} (x)$

Step 1

By the Sum Rule, the derivative of $\tan (x) + \frac{1}{3} \tan^{3} (x)$ with respect to $x$ is $\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [\frac{1}{3} \tan^{3} (x)]$ .

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{3} \tan^{3} (x)]$ .

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Step 3.1

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

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

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $\tan (x)$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $\tan (x)$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine $\frac{\tan^{2} (x) \cdot 3}{3}$ and $\sec^{2} (x)$ .

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

Step 3.7

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

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

Step 3.8

Cancel the common factor of $3$ .

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Step 3.8.1

Cancel the common factor.

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

Step 3.8.2

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

Factor $\sec^{2} (x)$ out of $\sec^{2} (x) \tan^{2} (x) + \sec^{2} (x)$ .

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Step 4.2.1

Factor $\sec^{2} (x)$ out of $\sec^{2} (x) \tan^{2} (x)$ .

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

Step 4.2.2

Multiply by $1$ .

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

Step 4.2.3

Factor $\sec^{2} (x)$ out of $\sec^{2} (x) (\tan^{2} (x)) + \sec^{2} (x) \cdot 1$ .

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

Step 4.3

Apply pythagorean identity.

$\sec^{2} (x) \sec^{2} (x)$

Step 4.4

Multiply $\sec^{2} (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 4.4.1

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

${\sec (x)}^{2 + 2}$

Step 4.4.2

Add $2$ and $2$ .

$\sec^{4} (x)$