Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.1.2

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

$\frac{1}{u} \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

$\frac{1}{y} \frac{d}{d x} [y]$

Step 2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{1}{y} y'$

Step 2.3

Combine $\frac{1}{y}$ and $y'$ .

$\frac{y'}{y}$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $f (x) = x$ and $g (x) = \ln (x)$ .

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

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.3.3

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

$1 + \ln (x) \cdot 1$

Step 3.3.4

Multiply $\ln (x)$ by $1$ .

$1 + \ln (x)$

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Multiply both sides by $y$ .

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

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Simplify the left side.

Tap for more steps...

Step 5.2.1.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.2.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

$y' = (1 + \ln (x)) y$

Step 5.2.2

Simplify the right side.

Tap for more steps...

Step 5.2.2.1

Simplify $(1 + \ln (x)) y$ .

Tap for more steps...

Step 5.2.2.1.1

Apply the distributive property.

$y' = 1 y + \ln (x) y$

Step 5.2.2.1.2

Simplify the expression.

Tap for more steps...

Step 5.2.2.1.2.1

Multiply $y$ by $1$ .

$y' = y + \ln (x) y$

Step 5.2.2.1.2.2

Reorder factors in $y + \ln (x) y$ .

$y' = y + y \ln (x)$

Step 6

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

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