Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

By the Sum Rule, the derivative of $x \ln (x) - x$ with respect to $x$ is $\frac{d}{d x} [x \ln (x)] + \frac{d}{d x} [- x]$ .

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.2.5.1

Cancel the common factor.

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

Step 3.2.5.2

Rewrite the expression.

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

Step 3.2.6

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

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

Step 3.3

Evaluate $\frac{d}{d x} [- x]$ .

Tap for more steps...

Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $- 1$ by $1$ .

$1 + \ln (x) - 1$

Step 3.4

Combine terms.

Tap for more steps...

Step 3.4.1

Subtract $1$ from $1$ .

$0 + \ln (x)$

Step 3.4.2

Add $0$ and $\ln (x)$ .

$\ln (x)$

Step 4

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

$y' = \ln (x)$

Step 5

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

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