Calculus Examples

$(5 - 2 x) e^{x}$

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 $f (x) = 5 - 2 x$ and $g (x) = e^{x}$ .

$(5 - 2 x) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [5 - 2 x]$

Step 2

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

$(5 - 2 x) e^{x} + e^{x} \frac{d}{d x} [5 - 2 x]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $5 - 2 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x]$ .

$(5 - 2 x) e^{x} + e^{x} (\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x])$

Step 3.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$(5 - 2 x) e^{x} + e^{x} (0 + \frac{d}{d x} [- 2 x])$

Step 3.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$(5 - 2 x) e^{x} + e^{x} \frac{d}{d x} [- 2 x]$

Step 3.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$(5 - 2 x) e^{x} + e^{x} (- 2 \frac{d}{d x} [x])$

Step 3.5

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

$(5 - 2 x) e^{x} + e^{x} (- 2 \cdot 1)$

Step 3.6

Simplify the expression.

Tap for more steps...

Step 3.6.1

Multiply $- 2$ by $1$ .

$(5 - 2 x) e^{x} + e^{x} \cdot - 2$

Step 3.6.2

Move $- 2$ to the left of $e^{x}$ .

$(5 - 2 x) e^{x} - 2 \cdot e^{x}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$5 e^{x} - 2 x e^{x} - 2 e^{x}$

Step 4.2

Subtract $2 e^{x}$ from $5 e^{x}$ .

$- 2 x e^{x} + 3 e^{x}$

Step 4.3

Reorder terms.

$- 2 e^{x} x + 3 e^{x}$

Step 4.4

Reorder factors in $- 2 e^{x} x + 3 e^{x}$ .

$- 2 x e^{x} + 3 e^{x}$