Calculus Examples

$y = \frac{\ln (x)}{4 x + 7}$

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 $g (x) = 4 x + 7$ .

$\frac{(4 x + 7) \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [4 x + 7]}{{(4 x + 7)}^{2}}$

Step 2

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

$\frac{(4 x + 7) \frac{1}{x} - \ln (x) \frac{d}{d x} [4 x + 7]}{{(4 x + 7)}^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x + 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [7]$ .

$\frac{(4 x + 7) \frac{1}{x} - \ln (x) (\frac{d}{d x} [4 x] + \frac{d}{d x} [7])}{{(4 x + 7)}^{2}}$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{(4 x + 7) \frac{1}{x} - \ln (x) (4 \frac{d}{d x} [x] + \frac{d}{d x} [7])}{{(4 x + 7)}^{2}}$

Step 3.3

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

$\frac{(4 x + 7) \frac{1}{x} - \ln (x) (4 \cdot 1 + \frac{d}{d x} [7])}{{(4 x + 7)}^{2}}$

Step 3.4

Multiply $4$ by $1$ .

$\frac{(4 x + 7) \frac{1}{x} - \ln (x) (4 + \frac{d}{d x} [7])}{{(4 x + 7)}^{2}}$

Step 3.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{(4 x + 7) \frac{1}{x} - \ln (x) (4 + 0)}{{(4 x + 7)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $4$ and $0$ .

$\frac{(4 x + 7) \frac{1}{x} - \ln (x) \cdot 4}{{(4 x + 7)}^{2}}$

Step 3.6.2

Multiply $4$ by $- 1$ .

$\frac{(4 x + 7) \frac{1}{x} - 4 \ln (x)}{{(4 x + 7)}^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{4 x \frac{1}{x} + 7 \frac{1}{x} - 4 \ln (x)}{{(4 x + 7)}^{2}}$

Step 4.2

Simplify each term.

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Step 4.2.1

Cancel the common factor of $x$ .

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Step 4.2.1.1

Factor $x$ out of $4 x$ .

$\frac{x \cdot 4 \frac{1}{x} + 7 \frac{1}{x} - 4 \ln (x)}{{(4 x + 7)}^{2}}$

Step 4.2.1.2

Cancel the common factor.

$\frac{x \cdot 4 \frac{1}{x} + 7 \frac{1}{x} - 4 \ln (x)}{{(4 x + 7)}^{2}}$

Step 4.2.1.3

Rewrite the expression.

$\frac{4 + 7 \frac{1}{x} - 4 \ln (x)}{{(4 x + 7)}^{2}}$

Step 4.2.2

Combine $7$ and $\frac{1}{x}$ .

$\frac{4 + \frac{7}{x} - 4 \ln (x)}{{(4 x + 7)}^{2}}$

Step 4.2.3

Simplify $- 4 \ln (x)$ by moving $4$ inside the logarithm.

$\frac{4 + \frac{7}{x} - \ln (x^{4})}{{(4 x + 7)}^{2}}$

Step 4.3

Combine terms.

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Step 4.3.1

Multiply $\frac{4 + \frac{7}{x} - \ln (x^{4})}{{(4 x + 7)}^{2}}$ by $\frac{x}{x}$ .

$\frac{x}{x} \cdot \frac{4 + \frac{7}{x} - \ln (x^{4})}{{(4 x + 7)}^{2}}$

Step 4.3.2

Combine.

$\frac{x (4 + \frac{7}{x} - \ln (x^{4}))}{x {(4 x + 7)}^{2}}$

Step 4.3.3

Apply the distributive property.

$\frac{x \cdot 4 + x \frac{7}{x} + x (- \ln (x^{4}))}{x {(4 x + 7)}^{2}}$

Step 4.3.4

Cancel the common factor of $x$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{x \cdot 4 + x \frac{7}{x} + x (- \ln (x^{4}))}{x {(4 x + 7)}^{2}}$

Step 4.3.4.2

Rewrite the expression.

$\frac{x \cdot 4 + 7 + x (- \ln (x^{4}))}{x {(4 x + 7)}^{2}}$

Step 4.3.5

Move $4$ to the left of $x$ .

$\frac{4 x + 7 + x (- \ln (x^{4}))}{x {(4 x + 7)}^{2}}$

Step 4.4

Reorder terms.

$\frac{- x \ln (x^{4}) + 4 x + 7}{x {(4 x + 7)}^{2}}$

Step 4.5

Factor $- 1$ out of $- x \ln (x^{4})$ .

$\frac{- (x \ln (x^{4})) + 4 x + 7}{x {(4 x + 7)}^{2}}$

Step 4.6

Factor $- 1$ out of $4 x$ .

$\frac{- (x \ln (x^{4})) - (- 4 x) + 7}{x {(4 x + 7)}^{2}}$

Step 4.7

Factor $- 1$ out of $- (x \ln (x^{4})) - (- 4 x)$ .

$\frac{- (x \ln (x^{4}) - 4 x) + 7}{x {(4 x + 7)}^{2}}$

Step 4.8

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{- (x \ln (x^{4}) - 4 x) - 1 (- 7)}{x {(4 x + 7)}^{2}}$

Step 4.9

Factor $- 1$ out of $- (x \ln (x^{4}) - 4 x) - 1 (- 7)$ .

$\frac{- (x \ln (x^{4}) - 4 x - 7)}{x {(4 x + 7)}^{2}}$

Step 4.10

Rewrite $- (x \ln (x^{4}) - 4 x - 7)$ as $- 1 (x \ln (x^{4}) - 4 x - 7)$ .

$\frac{- 1 (x \ln (x^{4}) - 4 x - 7)}{x {(4 x + 7)}^{2}}$

Step 4.11

Move the negative in front of the fraction.

$- \frac{x \ln (x^{4}) - 4 x - 7}{x {(4 x + 7)}^{2}}$