Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \ln (x)$ and $g (x) = x + 1$ .

$\frac{(x + 1) \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 2

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

$\frac{(x + 1) \frac{1}{x} - \ln (x) \frac{d}{d x} [x + 1]}{{(x + 1)}^{2}}$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{(x + 1) \frac{1}{x} - \ln (x) (\frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 3.2

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

$\frac{(x + 1) \frac{1}{x} - \ln (x) (1 + \frac{d}{d x} [1])}{{(x + 1)}^{2}}$

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(x + 1) \frac{1}{x} - \ln (x) (1 + 0)}{{(x + 1)}^{2}}$

Step 3.4

Simplify the expression.

Tap for more steps...

Step 3.4.1

Add $1$ and $0$ .

$\frac{(x + 1) \frac{1}{x} - \ln (x) \cdot 1}{{(x + 1)}^{2}}$

Step 3.4.2

Multiply $- 1$ by $1$ .

$\frac{(x + 1) \frac{1}{x} - \ln (x)}{{(x + 1)}^{2}}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$\frac{x \frac{1}{x} + 1 \frac{1}{x} - \ln (x)}{{(x + 1)}^{2}}$

Step 4.2

Simplify each term.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{x \frac{1}{x} + 1 \frac{1}{x} - \ln (x)}{{(x + 1)}^{2}}$

Step 4.2.1.2

Rewrite the expression.

$\frac{1 + 1 \frac{1}{x} - \ln (x)}{{(x + 1)}^{2}}$

Step 4.2.2

Multiply $\frac{1}{x}$ by $1$ .

$\frac{1 + \frac{1}{x} - \ln (x)}{{(x + 1)}^{2}}$

Step 4.3

Combine terms.

Tap for more steps...

Step 4.3.1

Multiply $\frac{1 + \frac{1}{x} - \ln (x)}{{(x + 1)}^{2}}$ by $\frac{x}{x}$ .

$\frac{x}{x} \cdot \frac{1 + \frac{1}{x} - \ln (x)}{{(x + 1)}^{2}}$

Step 4.3.2

Combine.

$\frac{x (1 + \frac{1}{x} - \ln (x))}{x {(x + 1)}^{2}}$

Step 4.3.3

Apply the distributive property.

$\frac{x \cdot 1 + x \frac{1}{x} + x (- \ln (x))}{x {(x + 1)}^{2}}$

Step 4.3.4

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.3.4.1

Cancel the common factor.

$\frac{x \cdot 1 + x \frac{1}{x} + x (- \ln (x))}{x {(x + 1)}^{2}}$

Step 4.3.4.2

Rewrite the expression.

$\frac{x \cdot 1 + 1 + x (- \ln (x))}{x {(x + 1)}^{2}}$

Step 4.3.5

Multiply $x$ by $1$ .

$\frac{x + 1 + x (- \ln (x))}{x {(x + 1)}^{2}}$

Step 4.4

Reorder terms.

$\frac{- x \ln (x) + x + 1}{x {(x + 1)}^{2}}$

Step 4.5

Factor $- 1$ out of $- x \ln (x)$ .

$\frac{- (x \ln (x)) + x + 1}{x {(x + 1)}^{2}}$

Step 4.6

Factor $- 1$ out of $x$ .

$\frac{- (x \ln (x)) - 1 (- x) + 1}{x {(x + 1)}^{2}}$

Step 4.7

Factor $- 1$ out of $- (x \ln (x)) - 1 (- x)$ .

$\frac{- (x \ln (x) - x) + 1}{x {(x + 1)}^{2}}$

Step 4.8

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (x \ln (x) - x) - 1 (- 1)}{x {(x + 1)}^{2}}$

Step 4.9

Factor $- 1$ out of $- (x \ln (x) - x) - 1 (- 1)$ .

$\frac{- (x \ln (x) - x - 1)}{x {(x + 1)}^{2}}$

Step 4.10

Rewrite $- (x \ln (x) - x - 1)$ as $- 1 (x \ln (x) - x - 1)$ .

$\frac{- 1 (x \ln (x) - x - 1)}{x {(x + 1)}^{2}}$

Step 4.11

Move the negative in front of the fraction.

$- \frac{x \ln (x) - x - 1}{x {(x + 1)}^{2}}$