Algebra Examples

$y = \frac{\log (x)}{4 + \log (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{\log (x)}{4 + \log (x)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = 4 + \log (x)$ .

$\frac{(4 + \log (x)) \frac{d}{d x} [\log (x)] - \log (x) \frac{d}{d x} [4 + \log (x)]}{{(4 + \log (x))}^{2}}$

Step 3.2

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

$\frac{(4 + \log (x)) \frac{1}{x \ln (10)} - \log (x) \frac{d}{d x} [4 + \log (x)]}{{(4 + \log (x))}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $4 + \log (x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [\log (x)]$ .

$\frac{(4 + \log (x)) \frac{1}{x \ln (10)} - \log (x) (\frac{d}{d x} [4] + \frac{d}{d x} [\log (x)])}{{(4 + \log (x))}^{2}}$

Step 3.3.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{(4 + \log (x)) \frac{1}{x \ln (10)} - \log (x) (0 + \frac{d}{d x} [\log (x)])}{{(4 + \log (x))}^{2}}$

Step 3.3.3

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

$\frac{(4 + \log (x)) \frac{1}{x \ln (10)} - \log (x) \frac{d}{d x} [\log (x)]}{{(4 + \log (x))}^{2}}$

Step 3.4

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

$\frac{(4 + \log (x)) \frac{1}{x \ln (10)} - \log (x) \frac{1}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.5

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

$\frac{(4 + \log (x)) \frac{1}{x \ln (10)} - \frac{\log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.6

To write $(4 + \log (x)) \frac{1}{x \ln (10)}$ as a fraction with a common denominator, multiply by $\frac{x \ln (10)}{x \ln (10)}$ .

$\frac{(4 + \log (x)) \frac{1}{x \ln (10)} \cdot \frac{x \ln (10)}{x \ln (10)} - \frac{\log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.7

Combine $(4 + \log (x)) \frac{1}{x \ln (10)}$ and $\frac{x \ln (10)}{x \ln (10)}$ .

$\frac{\frac{(4 + \log (x)) \frac{1}{x \ln (10)} (x \ln (10))}{x \ln (10)} - \frac{\log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.8

Combine the numerators over the common denominator.

$\frac{\frac{(4 + \log (x)) \frac{1}{x \ln (10)} (x \ln (10)) - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.9

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

$\frac{\frac{(4 + \log (x)) \frac{x}{x \ln (10)} \ln (10) - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.10

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

$\frac{\frac{(4 + \log (x)) \frac{\ln (10) x}{x \ln (10)} - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.11

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

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Step 3.11.1

Cancel the common factor.

$\frac{\frac{(4 + \log (x)) \frac{\ln (10) x}{x \ln (10)} - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.11.2

Rewrite the expression.

$\frac{\frac{(4 + \log (x)) \frac{x}{x} - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.12

Cancel the common factor of $x$ .

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Step 3.12.1

Cancel the common factor.

$\frac{\frac{(4 + \log (x)) \frac{x}{x} - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.12.2

Rewrite the expression.

$\frac{\frac{(4 + \log (x)) \cdot 1 - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.13

Multiply $4 + \log (x)$ by $1$ .

$\frac{\frac{4 + \log (x) - \log (x)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.14

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

$\frac{\frac{4 + \log (\frac{x}{x})}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.15

Cancel the common factor of $x$ .

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Step 3.15.1

Cancel the common factor.

$\frac{\frac{4 + \log (\frac{x}{x})}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.15.2

Rewrite the expression.

$\frac{\frac{4 + \log (1)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$

Step 3.16

Rewrite $\frac{\frac{4 + \log (1)}{x \ln (10)}}{{(4 + \log (x))}^{2}}$ as a product.

$\frac{4 + \log (1)}{x \ln (10)} \cdot \frac{1}{{(4 + \log (x))}^{2}}$

Step 3.17

Multiply $\frac{4 + \log (1)}{x \ln (10)}$ by $\frac{1}{{(4 + \log (x))}^{2}}$ .

$\frac{4 + \log (1)}{x \ln (10) {(4 + \log (x))}^{2}}$

Step 3.18

Simplify.

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Step 3.18.1

Simplify the numerator.

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Step 3.18.1.1

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

$\frac{4 + 0}{x \ln (10) {(4 + \log (x))}^{2}}$

Step 3.18.1.2

Add $4$ and $0$ .

$\frac{4}{x \ln (10) {(4 + \log (x))}^{2}}$

Step 3.18.2

Reorder terms.

$\frac{4}{x {(4 + \log (x))}^{2} \ln (10)}$

Step 3.18.3

Reorder factors in $\frac{4}{x {(4 + \log (x))}^{2} \ln (10)}$ .

$\frac{4}{x \ln (10) {(4 + \log (x))}^{2}}$

Step 4

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

$y' = \frac{4}{x \ln (10) {(4 + \log (x))}^{2}}$

Step 5

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

$\frac{d y}{d x} = \frac{4}{x \ln (10) {(4 + \log (x))}^{2}}$