Calculus Examples

${(\sqrt[4]{\sec (x)})}^{4}$

Step 1

Rewrite ${(\sqrt[4]{\sec (x)})}^{4}$ as $\sec (x)$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\sec (x)}$ as ${\sec (x)}^{\frac{1}{4}}$ .

$\frac{d}{d x} [{({\sec (x)}^{\frac{1}{4}})}^{4}]$

Step 1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [{\sec (x)}^{\frac{1}{4} \cdot 4}]$

Step 1.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 1.4

Cancel the common factor of $4$ .

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Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

$\frac{d}{d x} [\sec^{1} (x)]$

Step 1.5

Simplify.

$\frac{d}{d x} [\sec (x)]$

Step 2

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

$\sec (x) \tan (x)$