Calculus Examples

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

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 2 x$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $- 2 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $- 2 x$ .

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify the expression.

Tap for more steps...

Step 3.3.1

Multiply $- 2$ by $1$ .

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

Step 3.3.2

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Combine terms.

Tap for more steps...

Step 4.4.1

Multiply $e^{- 2 x}$ by $1$ .

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

Step 4.4.2

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

Tap for more steps...

Step 4.4.2.1

Move $e^{- 2 x}$ .

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

Step 4.4.2.2

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

$- 2 x^{2} e^{- 2 x} + e^{- 2 x} + 0$

Step 4.4.3

Add $- 2 x^{2} e^{- 2 x} + e^{- 2 x}$ and $0$ .

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

Step 4.5

Reorder terms.

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

Step 4.6

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

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