Calculus Examples

$y = e^{x} \sin^{5} (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{x} \sin^{5} (x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = e^{x}$ and $g (x) = \sin^{5} (x)$ .

$e^{x} \frac{d}{d x} [\sin^{5} (x)] + \sin^{5} (x) \frac{d}{d x} [e^{x}]$

Step 3.2

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

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Step 3.2.1

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

$e^{x} (\frac{d}{d u} [u^{5}] \frac{d}{d x} [\sin (x)]) + \sin^{5} (x) \frac{d}{d x} [e^{x}]$

Step 3.2.2

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

$e^{x} (5 u^{4} \frac{d}{d x} [\sin (x)]) + \sin^{5} (x) \frac{d}{d x} [e^{x}]$

Step 3.2.3

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

$e^{x} (5 \sin^{4} (x) \frac{d}{d x} [\sin (x)]) + \sin^{5} (x) \frac{d}{d x} [e^{x}]$

Step 3.3

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

$e^{x} (5 \sin^{4} (x) \cos (x)) + \sin^{5} (x) \frac{d}{d x} [e^{x}]$

Step 3.4

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

$e^{x} (5 \sin^{4} (x) \cos (x)) + \sin^{5} (x) e^{x}$

Step 3.5

Reorder terms.

$5 e^{x} \sin^{4} (x) \cos (x) + e^{x} \sin^{5} (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 5 e^{x} \sin^{4} (x) \cos (x) + e^{x} \sin^{5} (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 5 e^{x} \sin^{4} (x) \cos (x) + e^{x} \sin^{5} (x)$