Calculus Examples

$y = x^{5} e^{x}$

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.2

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

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

Step 3.3

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

$x^{5} e^{x} + e^{x} (5 x^{4})$

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

$e^{x} x^{5} + 5 e^{x} x^{4}$

Step 3.4.2

Reorder factors in $e^{x} x^{5} + 5 e^{x} x^{4}$ .

$x^{5} e^{x} + 5 x^{4} e^{x}$

Step 4

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

$y' = x^{5} e^{x} + 5 x^{4} e^{x}$

Step 5

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

$\frac{d y}{d x} = x^{5} e^{x} + 5 x^{4} e^{x}$