Calculus Examples

$g (x) = \sec (x^{4}) + \csc (x^{4})$

Step 1

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

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

Step 2

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

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

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{4}$ .

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

Step 2.1.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

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

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x^{4}$ .

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

Step 2.2

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

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u_{2}$ as $x^{4}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u_{2}$ with $x^{4}$ .

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

Step 3.2

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

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

Step 3.3

Multiply $4$ by $- 1$ .

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

Step 4

Reorder terms.

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