Calculus Examples

$f (x) = 3 \sin (x^{4}) e$

Step 1

Since $3 e$ is constant with respect to $x$ , the derivative of $3 \sin (x^{4}) e$ with respect to $x$ is $3 e \frac{d}{d x} [\sin (x^{4})]$ .

$3 e \frac{d}{d x} [\sin (x^{4})]$

Step 2

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

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Step 2.1

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

$3 e (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x^{4}])$

Step 2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$3 e (\cos (u) \frac{d}{d x} [x^{4}])$

Step 2.3

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

$3 e (\cos (x^{4}) \frac{d}{d x} [x^{4}])$

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 $n = 4$ .

$3 e \cos (x^{4}) (4 x^{3})$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $4$ by $3$ .

$12 e \cos (x^{4}) x^{3}$

Step 3.2.2

Reorder the factors of $12 e \cos (x^{4}) x^{3}$ .

$12 e x^{3} \cos (x^{4})$