Calculus Examples

$f (x) = 4 e^{x} \cos (x) - \sin (x) \cdot (4 e^{x})$

Step 1

By the Sum Rule, the derivative of $4 e^{x} \cos (x) - \sin (x) \cdot (4 e^{x})$ with respect to $x$ is $\frac{d}{d x} [4 e^{x} \cos (x)] + \frac{d}{d x} [- \sin (x) \cdot (4 e^{x})]$ .

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Evaluate $\frac{d}{d x} [- \sin (x) \cdot (4 e^{x})]$ .

Tap for more steps...

Step 3.1

Multiply $4$ by $- 1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Combine terms.

Tap for more steps...

Step 4.3.1

Multiply $- 1$ by $4$ .

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

Step 4.3.2

Subtract $4 \sin (x) e^{x}$ from $- 4 e^{x} \sin (x)$ .

Tap for more steps...

Step 4.3.2.1

Move $\sin (x)$ .

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

Step 4.3.2.2

Subtract $4 e^{x} \sin (x)$ from $- 4 e^{x} \sin (x)$ .

$- 8 e^{x} \sin (x) + 4 \cos (x) e^{x} - 4 e^{x} \cos (x)$

Step 4.3.3

Subtract $4 e^{x} \cos (x)$ from $4 \cos (x) e^{x}$ .

Tap for more steps...

Step 4.3.3.1

Move $\cos (x)$ .

$- 8 e^{x} \sin (x) + 4 e^{x} \cos (x) - 4 e^{x} \cos (x)$

Step 4.3.3.2

Subtract $4 e^{x} \cos (x)$ from $4 e^{x} \cos (x)$ .

$- 8 e^{x} \sin (x) + 0$

Step 4.3.4

Add $- 8 e^{x} \sin (x)$ and $0$ .

$- 8 e^{x} \sin (x)$