Calculus Examples

$y = \log (\ln (x))$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = \ln (x)$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

$\frac{d}{d u} [\log (u)] \frac{d}{d x} [\ln (x)]$

Step 3.1.2

The derivative of $\log (u)$ with respect to $u$ is $\frac{1}{u \ln (10)}$ .

$\frac{1}{u \ln (10)} \frac{d}{d x} [\ln (x)]$

Step 3.1.3

Replace all occurrences of $u$ with $\ln (x)$ .

$\frac{1}{\ln (x) \ln (10)} \frac{d}{d x} [\ln (x)]$

Step 3.2

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

$\frac{1}{\ln (x) \ln (10)} \cdot \frac{1}{x}$

Step 3.3

Combine fractions.

Tap for more steps...

Step 3.3.1

Multiply $\frac{1}{\ln (x) \ln (10)}$ by $\frac{1}{x}$ .

$\frac{1}{\ln (x) \ln (10) x}$

Step 3.3.2

Reorder terms.

$\frac{1}{x \ln (10) \ln (x)}$

Step 4

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

$y' = \frac{1}{x \ln (10) \ln (x)}$

Step 5

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

$\frac{d y}{d x} = \frac{1}{x \ln (10) \ln (x)}$