Calculus Examples

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

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

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

Step 2

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

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

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

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

Step 4

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

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

Step 5

Differentiate using the Power Rule.

Tap for more steps...

Step 5.1

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

$x e^{x} \cos (x) + \sin (x) (x e^{x} + e^{x} \cdot 1)$

Step 5.2

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

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Reorder terms.

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