Calculus Examples

$y = \frac{x^{2}}{e^{x}}$

Step 1

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 $g (x) = e^{x}$ .

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.2

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

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

Step 4.2

Reorder terms.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Multiply by $1$ .

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

Step 4.4.2.2

Cancel the common factor.

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

Step 4.4.2.3

Rewrite the expression.

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

Step 4.4.2.4

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

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

Step 4.5

Apply the distributive property.

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

Step 4.6

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

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

Step 4.7

Rewrite using the commutative property of multiplication.

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

Step 4.8

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

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Step 4.8.1

Move $x$ .

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

Step 4.8.2

Multiply $x$ by $x$ .

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

Step 4.9

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

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