Calculus Examples

$\sec^{4} (x) - \tan^{4} (x)$

Step 1

By the Sum Rule, the derivative of $\sec^{4} (x) - \tan^{4} (x)$ with respect to $x$ is $\frac{d}{d x} [\sec^{4} (x)] + \frac{d}{d x} [- \tan^{4} (x)]$ .

$\frac{d}{d x} [\sec^{4} (x)] + \frac{d}{d x} [- \tan^{4} (x)]$

Step 2

Evaluate $\frac{d}{d x} [\sec^{4} (x)]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \sec (x)$ .

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Step 2.1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- \tan^{4} (x)]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 4$ .

$4 {u_{1}}^{3} \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- \tan^{4} (x)]$

Step 2.1.3

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

$4 \sec^{3} (x) \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- \tan^{4} (x)]$

Step 2.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$4 \sec^{3} (x) (\sec (x) \tan (x)) + \frac{d}{d x} [- \tan^{4} (x)]$

Step 2.3

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

$4 (\sec^{1} (x) \sec^{3} (x)) \tan (x) + \frac{d}{d x} [- \tan^{4} (x)]$

Step 2.4

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

$4 {\sec (x)}^{1 + 3} \tan (x) + \frac{d}{d x} [- \tan^{4} (x)]$

Step 2.5

Add $1$ and $3$ .

$4 \sec^{4} (x) \tan (x) + \frac{d}{d x} [- \tan^{4} (x)]$

Step 3

Evaluate $\frac{d}{d x} [- \tan^{4} (x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \tan^{4} (x)$ with respect to $x$ is $- \frac{d}{d x} [\tan^{4} (x)]$ .

$4 \sec^{4} (x) \tan (x) - \frac{d}{d x} [\tan^{4} (x)]$

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 $f (x) = x^{4}$ and $g (x) = \tan (x)$ .

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Step 3.2.1

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

$4 \sec^{4} (x) \tan (x) - (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [\tan (x)])$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$4 \sec^{4} (x) \tan (x) - (4 {u_{2}}^{3} \frac{d}{d x} [\tan (x)])$

Step 3.2.3

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

$4 \sec^{4} (x) \tan (x) - (4 \tan^{3} (x) \frac{d}{d x} [\tan (x)])$

Step 3.3

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

$4 \sec^{4} (x) \tan (x) - (4 \tan^{3} (x) \sec^{2} (x))$

Step 3.4

Multiply $4$ by $- 1$ .

$4 \sec^{4} (x) \tan (x) - 4 \tan^{3} (x) \sec^{2} (x)$

Step 4

Simplify.

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Step 4.1

Factor $4 \sec^{2} (x) \tan (x)$ out of $4 \sec^{4} (x) \tan (x) - 4 \tan^{3} (x) \sec^{2} (x)$ .

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Step 4.1.1

Factor $4 \sec^{2} (x) \tan (x)$ out of $4 \sec^{4} (x) \tan (x)$ .

$4 \sec^{2} (x) \tan (x) (\sec^{2} (x)) - 4 \tan^{3} (x) \sec^{2} (x)$

Step 4.1.2

Factor $4 \sec^{2} (x) \tan (x)$ out of $- 4 \tan^{3} (x) \sec^{2} (x)$ .

$4 \sec^{2} (x) \tan (x) (\sec^{2} (x)) + 4 \sec^{2} (x) \tan (x) (- \tan^{2} (x))$

Step 4.1.3

Factor $4 \sec^{2} (x) \tan (x)$ out of $4 \sec^{2} (x) \tan (x) (\sec^{2} (x)) + 4 \sec^{2} (x) \tan (x) (- \tan^{2} (x))$ .

$4 \sec^{2} (x) \tan (x) (\sec^{2} (x) - \tan^{2} (x))$

Step 4.2

Apply pythagorean identity.

$4 \sec^{2} (x) \tan (x) \cdot 1$

Step 4.3

Multiply $4$ by $1$ .

$4 \sec^{2} (x) \tan (x)$