Calculus Examples

$(x^{2} + 3 x - 9) e^{x}$

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 $f (x) = x^{2} + 3 x - 9$ and $g (x) = e^{x}$ .

$(x^{2} + 3 x - 9) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2} + 3 x - 9]$

Step 2

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

$(x^{2} + 3 x - 9) e^{x} + e^{x} \frac{d}{d x} [x^{2} + 3 x - 9]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $x^{2} + 3 x - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$ .

$(x^{2} + 3 x - 9) e^{x} + e^{x} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(x^{2} + 3 x - 9) e^{x} + e^{x} (2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9])$

Step 3.3

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

$(x^{2} + 3 x - 9) e^{x} + e^{x} (2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9])$

Step 3.4

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

$(x^{2} + 3 x - 9) e^{x} + e^{x} (2 x + 3 \cdot 1 + \frac{d}{d x} [- 9])$

Step 3.5

Multiply $3$ by $1$ .

$(x^{2} + 3 x - 9) e^{x} + e^{x} (2 x + 3 + \frac{d}{d x} [- 9])$

Step 3.6

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$(x^{2} + 3 x - 9) e^{x} + e^{x} (2 x + 3 + 0)$

Step 3.7

Add $2 x + 3$ and $0$ .

$(x^{2} + 3 x - 9) e^{x} + e^{x} (2 x + 3)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$x^{2} e^{x} + 3 x e^{x} - 9 e^{x} + e^{x} (2 x + 3)$

Step 4.2

Apply the distributive property.

$x^{2} e^{x} + 3 x e^{x} - 9 e^{x} + e^{x} (2 x) + e^{x} \cdot 3$

Step 4.3

Combine terms.

Tap for more steps...

Step 4.3.1

Move $3$ to the left of $e^{x}$ .

$x^{2} e^{x} + 3 x e^{x} - 9 e^{x} + e^{x} (2 x) + 3 \cdot e^{x}$

Step 4.3.2

Add $3 x e^{x}$ and $e^{x} (2 x)$ .

Tap for more steps...

Step 4.3.2.1

Move $e^{x}$ .

$x^{2} e^{x} + 3 x e^{x} + 2 x e^{x} - 9 e^{x} + 3 e^{x}$

Step 4.3.2.2

Add $3 x e^{x}$ and $2 x e^{x}$ .

$x^{2} e^{x} + 5 x e^{x} - 9 e^{x} + 3 e^{x}$

Step 4.3.3

Add $- 9 e^{x}$ and $3 e^{x}$ .

$x^{2} e^{x} + 5 x e^{x} - 6 e^{x}$

Step 4.4

Reorder terms.

$e^{x} x^{2} + 5 e^{x} x - 6 e^{x}$

Step 4.5

Reorder factors in $e^{x} x^{2} + 5 e^{x} x - 6 e^{x}$ .

$x^{2} e^{x} + 5 x e^{x} - 6 e^{x}$