Calculus Examples

$y = \frac{\ln (x)}{e^{x}}$

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 $f (x) = \ln (x)$ and $g (x) = e^{x}$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 4

Combine $e^{x}$ and $\frac{1}{x}$ .

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

Step 5

Multiply $\frac{\frac{e^{x}}{x} - \ln (x) \frac{d}{d x} [e^{x}]}{e^{2 x}}$ by $\frac{x}{x}$ .

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

Step 6

Simplify terms.

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Step 6.1

Combine.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Cancel the common factor of $x$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Rewrite using the commutative property of multiplication.

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

Step 8.2

Reorder terms.

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

Multiply by $1$ .

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

Step 8.3.3

Factor $e^{x}$ out of $e^{x} (- x \ln (x)) + e^{x} \cdot 1$ .

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

Step 8.4

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

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Step 8.4.1

Factor $e^{2 x}$ out of $e^{x} (- x \ln (x) + 1)$ .

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

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

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

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

Step 8.4.2.2

Cancel the common factor.

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

Step 8.4.2.3

Rewrite the expression.

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

Step 8.5

Factor $- 1$ out of $- x \ln (x)$ .

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

Step 8.6

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 8.7

Factor $- 1$ out of $- (x \ln (x)) - 1 (- 1)$ .

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

Step 8.8

Rewrite $- (x \ln (x) - 1)$ as $- 1 (x \ln (x) - 1)$ .

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

Step 8.9

Move the negative in front of the fraction.

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