Calculus Examples

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

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

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

Step 2

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

$\frac{(1 + \log (x)) \frac{1}{x \ln (10)} - \log (x) \frac{d}{d x} [1 + \log (x)]}{{(1 + \log (x))}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$\frac{(1 + \log (x)) \frac{1}{x \ln (10)} - \log (x) (\frac{d}{d x} [1] + \frac{d}{d x} [\log (x)])}{{(1 + \log (x))}^{2}}$

Step 3.2

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

$\frac{(1 + \log (x)) \frac{1}{x \ln (10)} - \log (x) (0 + \frac{d}{d x} [\log (x)])}{{(1 + \log (x))}^{2}}$

Step 3.3

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

$\frac{(1 + \log (x)) \frac{1}{x \ln (10)} - \log (x) \frac{d}{d x} [\log (x)]}{{(1 + \log (x))}^{2}}$

Step 4

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

$\frac{(1 + \log (x)) \frac{1}{x \ln (10)} - \log (x) \frac{1}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 5

Combine $\frac{1}{x \ln (10)}$ and $\log (x)$ .

$\frac{(1 + \log (x)) \frac{1}{x \ln (10)} - \frac{\log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 6

To write $(1 + \log (x)) \frac{1}{x \ln (10)}$ as a fraction with a common denominator, multiply by $\frac{x \ln (10)}{x \ln (10)}$ .

$\frac{(1 + \log (x)) \frac{1}{x \ln (10)} \cdot \frac{x \ln (10)}{x \ln (10)} - \frac{\log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 7

Combine $(1 + \log (x)) \frac{1}{x \ln (10)}$ and $\frac{x \ln (10)}{x \ln (10)}$ .

$\frac{\frac{(1 + \log (x)) \frac{1}{x \ln (10)} (x \ln (10))}{x \ln (10)} - \frac{\log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 8

Combine the numerators over the common denominator.

$\frac{\frac{(1 + \log (x)) \frac{1}{x \ln (10)} (x \ln (10)) - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 9

Combine $x$ and $\frac{1}{x \ln (10)}$ .

$\frac{\frac{(1 + \log (x)) \frac{x}{x \ln (10)} \ln (10) - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 10

Combine $\ln (10)$ and $\frac{x}{x \ln (10)}$ .

$\frac{\frac{(1 + \log (x)) \frac{\ln (10) x}{x \ln (10)} - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 11

Cancel the common factor of $\ln (10)$ .

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Step 11.1

Cancel the common factor.

$\frac{\frac{(1 + \log (x)) \frac{\ln (10) x}{x \ln (10)} - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 11.2

Rewrite the expression.

$\frac{\frac{(1 + \log (x)) \frac{x}{x} - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 12

Cancel the common factor of $x$ .

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Step 12.1

Cancel the common factor.

$\frac{\frac{(1 + \log (x)) \frac{x}{x} - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 12.2

Rewrite the expression.

$\frac{\frac{(1 + \log (x)) \cdot 1 - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 13

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

$\frac{\frac{1 + \log (x) - \log (x)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 14

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\frac{1 + \log (\frac{x}{x})}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 15

Cancel the common factor of $x$ .

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Step 15.1

Cancel the common factor.

$\frac{\frac{1 + \log (\frac{x}{x})}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 15.2

Rewrite the expression.

$\frac{\frac{1 + \log (1)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$

Step 16

Rewrite $\frac{\frac{1 + \log (1)}{x \ln (10)}}{{(1 + \log (x))}^{2}}$ as a product.

$\frac{1 + \log (1)}{x \ln (10)} \cdot \frac{1}{{(1 + \log (x))}^{2}}$

Step 17

Multiply $\frac{1 + \log (1)}{x \ln (10)}$ by $\frac{1}{{(1 + \log (x))}^{2}}$ .

$\frac{1 + \log (1)}{x \ln (10) {(1 + \log (x))}^{2}}$

Step 18

Simplify.

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Step 18.1

Simplify the numerator.

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Step 18.1.1

Logarithm base $10$ of $1$ is $0$ .

$\frac{1 + 0}{x \ln (10) {(1 + \log (x))}^{2}}$

Step 18.1.2

Add $1$ and $0$ .

$\frac{1}{x \ln (10) {(1 + \log (x))}^{2}}$

Step 18.2

Reorder terms.

$\frac{1}{x {(1 + \log (x))}^{2} \ln (10)}$

Step 18.3

Reorder factors in $\frac{1}{x {(1 + \log (x))}^{2} \ln (10)}$ .

$\frac{1}{x \ln (10) {(1 + \log (x))}^{2}}$