Calculus Examples

$f (x) = \frac{2 (x^{2})}{e^{x}}$

Step 1

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

$2 \frac{d}{d x} [\frac{x^{2}}{e^{x}}]$

Step 2

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

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

Step 3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.1

Multiply the exponents in ${(e^{x})}^{2}$ .

Tap for more steps...

Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2

Move $2$ to the left of $x$ .

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

Step 3.2

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

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

Step 4

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

$2 \frac{e^{x} (2 x) - x^{2} e^{x}}{e^{2 x}}$

Step 5

Combine $2$ and $\frac{e^{x} (2 x) - x^{2} e^{x}}{e^{2 x}}$ .

$\frac{2 (e^{x} (2 x) - x^{2} e^{x})}{e^{2 x}}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Simplify the numerator.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.2.1.2

Multiply $2$ by $2$ .

$\frac{4 (e^{x} x) + 2 (- x^{2} e^{x})}{e^{2 x}}$

Step 6.2.1.3

Multiply $- 1$ by $2$ .

$\frac{4 e^{x} x - 2 x^{2} e^{x}}{e^{2 x}}$

Step 6.2.2

Reorder factors in $4 e^{x} x - 2 x^{2} e^{x}$ .

$\frac{4 x e^{x} - 2 x^{2} e^{x}}{e^{2 x}}$

Step 6.3

Reorder terms.

$\frac{4 e^{x} x - 2 e^{x} x^{2}}{e^{2 x}}$

Step 6.4

Factor $2 e^{x} x$ out of $4 e^{x} x - 2 e^{x} x^{2}$ .

Tap for more steps...

Step 6.4.1

Factor $2 e^{x} x$ out of $4 e^{x} x$ .

$\frac{2 e^{x} x (2) - 2 e^{x} x^{2}}{e^{2 x}}$

Step 6.4.2

Factor $2 e^{x} x$ out of $- 2 e^{x} x^{2}$ .

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

Step 6.4.3

Factor $2 e^{x} x$ out of $2 e^{x} x (2) + 2 e^{x} x (- x)$ .

$\frac{2 e^{x} x (2 - x)}{e^{2 x}}$

Step 6.5

Cancel the common factor of $e^{x}$ and $e^{2 x}$ .

Tap for more steps...

Step 6.5.1

Factor $e^{2 x}$ out of $2 e^{x} x (2 - x)$ .

$\frac{e^{2 x} (2 e^{- x} x (2 - x))}{e^{2 x}}$

Step 6.5.2

Cancel the common factors.

Tap for more steps...

Step 6.5.2.1

Multiply by $1$ .

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

Step 6.5.2.2

Cancel the common factor.

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

Step 6.5.2.3

Rewrite the expression.

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

Step 6.5.2.4

Divide $2 e^{- x} x (2 - x)$ by $1$ .

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

Step 6.6

Apply the distributive property.

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

Step 6.7

Multiply $2$ by $2$ .

$4 e^{- x} x + 2 e^{- x} x (- x)$

Step 6.8

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 6.8.1

Move $x$ .

$4 e^{- x} x + 2 e^{- x} (x \cdot x) \cdot - 1$

Step 6.8.2

Multiply $x$ by $x$ .

$4 e^{- x} x + 2 e^{- x} x^{2} \cdot - 1$

Step 6.9

Multiply $- 1$ by $2$ .

$4 e^{- x} x - 2 e^{- x} x^{2}$

Step 6.10

Reorder factors in $4 e^{- x} x - 2 e^{- x} x^{2}$ .

$4 x e^{- x} - 2 x^{2} e^{- x}$