Calculus Examples

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

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 $f (x) = x$ and $g (x) = 1 + \ln (x)$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $0$ and $\frac{d}{d x} [\ln (x)]$ .

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

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

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

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $- 1$ by $1$ .

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

Step 4.3.2

Subtract $1$ from $1$ .

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

Step 4.3.3

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

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