Calculus Examples

$e^{x \tan (x)}$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [x \tan (x)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [x \tan (x)]$

Step 1.3

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \tan (x)$ .

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Multiply $\tan (x)$ by $1$ .

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

Step 5

Apply the distributive property.

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