Calculus Examples

$f (x) = \tan^{5} (x^{5})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \tan (x^{5})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\tan (x^{5})$ .

$\frac{d}{d u_{1}} [{u_{1}}^{5}] \frac{d}{d x} [\tan (x^{5})]$

Step 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 $n = 5$ .

$5 {u_{1}}^{4} \frac{d}{d x} [\tan (x^{5})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\tan (x^{5})$ .

$5 \tan^{4} (x^{5}) \frac{d}{d x} [\tan (x^{5})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x^{5}$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x^{5}$ .

$5 \tan^{4} (x^{5}) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d x} [x^{5}])$

Step 2.2

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

$5 \tan^{4} (x^{5}) (\sec^{2} (u_{2}) \frac{d}{d x} [x^{5}])$

Step 2.3

Replace all occurrences of $u_{2}$ with $x^{5}$ .

$5 \tan^{4} (x^{5}) (\sec^{2} (x^{5}) \frac{d}{d x} [x^{5}])$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 \tan^{4} (x^{5}) \sec^{2} (x^{5}) (5 x^{4})$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $5$ by $5$ .

$25 \tan^{4} (x^{5}) \sec^{2} (x^{5}) x^{4}$

Step 3.2.2

Reorder the factors of $25 \tan^{4} (x^{5}) \sec^{2} (x^{5}) x^{4}$ .

$25 x^{4} \tan^{4} (x^{5}) \sec^{2} (x^{5})$