Calculus Examples

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

Step 1

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

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

Step 2

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 $g (x) = \cos (x)$ .

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

Step 3

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

$3 (e^{x} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{x}])$

Step 4

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

$3 (e^{x} (- \sin (x)) + \cos (x) e^{x})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$3 (e^{x} (- \sin (x))) + 3 (\cos (x) e^{x})$

Step 5.2

Multiply $- 1$ by $3$ .

$- 3 e^{x} \sin (x) + 3 \cos (x) e^{x}$

Step 5.3

Reorder terms.

$- 3 e^{x} \sin (x) + 3 e^{x} \cos (x)$