Calculus Examples

$y = \tan (15 x e^{x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 15 x e^{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $15 x e^{x}$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [15 x e^{x}]$

Step 1.2

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

$\sec^{2} (u) \frac{d}{d x} [15 x e^{x}]$

Step 1.3

Replace all occurrences of $u$ with $15 x e^{x}$ .

$\sec^{2} (15 x e^{x}) \frac{d}{d x} [15 x e^{x}]$

Step 2

Since $15$ is constant with respect to $x$ , the derivative of $15 x e^{x}$ with respect to $x$ is $15 \frac{d}{d x} [x e^{x}]$ .

$\sec^{2} (15 x e^{x}) (15 \frac{d}{d x} [x e^{x}])$

Step 3

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 $g (x) = e^{x}$ .

$\sec^{2} (15 x e^{x}) (15 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]))$

Step 4

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

$\sec^{2} (15 x e^{x}) (15 (x e^{x} + e^{x} \frac{d}{d x} [x]))$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

$\sec^{2} (15 x e^{x}) (15 (x e^{x} + e^{x} \cdot 1))$

Step 5.2

Multiply $e^{x}$ by $1$ .

$\sec^{2} (15 x e^{x}) (15 (x e^{x} + e^{x}))$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\sec^{2} (15 x e^{x}) (15 (x e^{x}) + 15 e^{x})$

Step 6.2

Reorder terms.

$\sec^{2} (15 x e^{x}) (15 e^{x} x + 15 e^{x})$

Step 6.3

Reorder factors in $15 e^{x} x + 15 e^{x}$ .

$\sec^{2} (15 x e^{x}) (15 x e^{x} + 15 e^{x})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\sec^{2} (15 x e^{x}) (15 x e^{x}) + \sec^{2} (15 x e^{x}) (15 e^{x})$

Step 7.2

Reorder terms.

$15 e^{x} x \sec^{2} (15 x e^{x}) + 15 e^{x} \sec^{2} (15 x e^{x})$