Calculus Examples

$y = \ln (e^{x^{2}})$

Step 1

Differentiate both sides of the equation.

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

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

Use logarithm rules to move $x^{2}$ out of the exponent.

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

Step 3.2

The natural logarithm of $e$ is $1$ .

$\frac{d}{d x} [x^{2} \cdot 1]$

Step 3.3

Multiply $x^{2}$ by $1$ .

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

Step 3.4

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

$2 x$

Step 4

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

$y' = 2 x$

Step 5

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

$\frac{d y}{d x} = 2 x$