Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify.

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Step 3.5.1

Reorder factors in $\ln (x) e^{x} - \frac{e^{x}}{x}$ .

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

Step 3.5.2

Simplify the numerator.

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Step 3.5.2.1

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

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Step 3.5.2.1.1

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

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

Step 3.5.2.1.2

Factor $e^{x}$ out of $- \frac{e^{x}}{x}$ .

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

Step 3.5.2.1.3

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

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

Step 3.5.2.2

To write $\ln (x)$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.5.2.3

Combine the numerators over the common denominator.

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

Step 3.5.3

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

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

Step 3.5.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5.5

Combine.

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

Step 3.5.6

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

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

Step 3.5.7

Reorder factors in $\frac{e^{x} (\ln (x) x - 1)}{x \ln^{2} (x)}$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{e^{x} (x \ln (x) - 1)}{x \ln^{2} (x)}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{e^{x} (x \ln (x) - 1)}{x \ln^{2} (x)}$