Algebra Examples

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

Differentiate both sides of the equation.

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

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

$y'$

Differentiate the right side of the equation.

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

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

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

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

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

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

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

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

Reorder terms.

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

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

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

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

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