Calculus Examples

$f (x) = \ln (\frac{x^{2}}{4 x - 3})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{x^{2}}{4 x - 3}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x^{2}}{4 x - 3}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{x^{2}}{4 x - 3}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [\frac{x^{2}}{4 x - 3}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x^{2}}{4 x - 3}$ .

$\frac{1}{\frac{x^{2}}{4 x - 3}} \frac{d}{d x} [\frac{x^{2}}{4 x - 3}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{x^{2}}{4 x - 3}$ .

$1 \frac{4 x - 3}{x^{2}} \frac{d}{d x} [\frac{x^{2}}{4 x - 3}]$

Step 3

Multiply $\frac{4 x - 3}{x^{2}}$ by $1$ .

$\frac{4 x - 3}{x^{2}} \frac{d}{d x} [\frac{x^{2}}{4 x - 3}]$

Step 4

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 $f (x) = x^{2}$ and $g (x) = 4 x - 3$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{(4 x - 3) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [4 x - 3]}{{(4 x - 3)}^{2}}$

Step 5

Differentiate.

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Step 5.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{(4 x - 3) (2 x) - x^{2} \frac{d}{d x} [4 x - 3]}{{(4 x - 3)}^{2}}$

Step 5.2

Move $2$ to the left of $4 x - 3$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 \cdot (4 x - 3) x - x^{2} \frac{d}{d x} [4 x - 3]}{{(4 x - 3)}^{2}}$

Step 5.3

By the Sum Rule, the derivative of $4 x - 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3]$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 (4 x - 3) x - x^{2} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 3])}{{(4 x - 3)}^{2}}$

Step 5.4

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

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 (4 x - 3) x - x^{2} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])}{{(4 x - 3)}^{2}}$

Step 5.5

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

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 (4 x - 3) x - x^{2} (4 \cdot 1 + \frac{d}{d x} [- 3])}{{(4 x - 3)}^{2}}$

Step 5.6

Multiply $4$ by $1$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 (4 x - 3) x - x^{2} (4 + \frac{d}{d x} [- 3])}{{(4 x - 3)}^{2}}$

Step 5.7

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 (4 x - 3) x - x^{2} (4 + 0)}{{(4 x - 3)}^{2}}$

Step 5.8

Combine fractions.

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Step 5.8.1

Add $4$ and $0$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 (4 x - 3) x - x^{2} \cdot 4}{{(4 x - 3)}^{2}}$

Step 5.8.2

Multiply $4$ by $- 1$ .

$\frac{4 x - 3}{x^{2}} \cdot \frac{2 (4 x - 3) x - 4 x^{2}}{{(4 x - 3)}^{2}}$

Step 5.8.3

Multiply $\frac{4 x - 3}{x^{2}}$ by $\frac{2 (4 x - 3) x - 4 x^{2}}{{(4 x - 3)}^{2}}$ .

$\frac{(4 x - 3) (2 (4 x - 3) x - 4 x^{2})}{x^{2} {(4 x - 3)}^{2}}$

Step 6

Cancel the common factors.

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Step 6.1

Factor $4 x - 3$ out of $x^{2} {(4 x - 3)}^{2}$ .

$\frac{(4 x - 3) (2 (4 x - 3) x - 4 x^{2})}{(4 x - 3) (x^{2} (4 x - 3))}$

Step 6.2

Cancel the common factor.

$\frac{(4 x - 3) (2 (4 x - 3) x - 4 x^{2})}{(4 x - 3) (x^{2} (4 x - 3))}$

Step 6.3

Rewrite the expression.

$\frac{2 (4 x - 3) x - 4 x^{2}}{x^{2} (4 x - 3)}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{(2 (4 x) + 2 \cdot - 3) x - 4 x^{2}}{x^{2} (4 x - 3)}$

Step 7.2

Apply the distributive property.

$\frac{2 (4 x) x + 2 \cdot - 3 x - 4 x^{2}}{x^{2} (4 x - 3)}$

Step 7.3

Apply the distributive property.

$\frac{2 (4 x) x + 2 \cdot - 3 x - 4 x^{2}}{x^{2} (4 x) + x^{2} \cdot - 3}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Simplify each term.

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Step 7.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 7.4.1.1.1

Move $x$ .

$\frac{2 (4 (x \cdot x)) + 2 \cdot - 3 x - 4 x^{2}}{x^{2} (4 x) + x^{2} \cdot - 3}$

Step 7.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 (4 x^{2}) + 2 \cdot - 3 x - 4 x^{2}}{x^{2} (4 x) + x^{2} \cdot - 3}$

Step 7.4.1.2

Multiply $4$ by $2$ .

$\frac{8 x^{2} + 2 \cdot - 3 x - 4 x^{2}}{x^{2} (4 x) + x^{2} \cdot - 3}$

Step 7.4.1.3

Multiply $2$ by $- 3$ .

$\frac{8 x^{2} - 6 x - 4 x^{2}}{x^{2} (4 x) + x^{2} \cdot - 3}$

Step 7.4.2

Subtract $4 x^{2}$ from $8 x^{2}$ .

$\frac{4 x^{2} - 6 x}{x^{2} (4 x) + x^{2} \cdot - 3}$

Step 7.5

Combine terms.

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Step 7.5.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.5.1.1

Move $x$ .

$\frac{4 x^{2} - 6 x}{x \cdot x^{2} \cdot 4 + x^{2} \cdot - 3}$

Step 7.5.1.2

Multiply $x$ by $x^{2}$ .

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Step 7.5.1.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{2} - 6 x}{x^{1} x^{2} \cdot 4 + x^{2} \cdot - 3}$

Step 7.5.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{2} - 6 x}{x^{1 + 2} \cdot 4 + x^{2} \cdot - 3}$

Step 7.5.1.3

Add $1$ and $2$ .

$\frac{4 x^{2} - 6 x}{x^{3} \cdot 4 + x^{2} \cdot - 3}$

Step 7.5.2

Move $4$ to the left of $x^{3}$ .

$\frac{4 x^{2} - 6 x}{4 \cdot x^{3} + x^{2} \cdot - 3}$

Step 7.5.3

Move $- 3$ to the left of $x^{2}$ .

$\frac{4 x^{2} - 6 x}{4 x^{3} - 3 x^{2}}$

Step 7.6

Factor $2 x$ out of $4 x^{2} - 6 x$ .

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Step 7.6.1

Factor $2 x$ out of $4 x^{2}$ .

$\frac{2 x (2 x) - 6 x}{4 x^{3} - 3 x^{2}}$

Step 7.6.2

Factor $2 x$ out of $- 6 x$ .

$\frac{2 x (2 x) + 2 x (- 3)}{4 x^{3} - 3 x^{2}}$

Step 7.6.3

Factor $2 x$ out of $2 x (2 x) + 2 x (- 3)$ .

$\frac{2 x (2 x - 3)}{4 x^{3} - 3 x^{2}}$

Step 7.7

Factor $x^{2}$ out of $4 x^{3} - 3 x^{2}$ .

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Step 7.7.1

Factor $x^{2}$ out of $4 x^{3}$ .

$\frac{2 x (2 x - 3)}{x^{2} (4 x) - 3 x^{2}}$

Step 7.7.2

Factor $x^{2}$ out of $- 3 x^{2}$ .

$\frac{2 x (2 x - 3)}{x^{2} (4 x) + x^{2} \cdot - 3}$

Step 7.7.3

Factor $x^{2}$ out of $x^{2} (4 x) + x^{2} \cdot - 3$ .

$\frac{2 x (2 x - 3)}{x^{2} (4 x - 3)}$

Step 7.8

Cancel the common factor of $x$ and $x^{2}$ .

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Step 7.8.1

Factor $x$ out of $2 x (2 x - 3)$ .

$\frac{x (2 (2 x - 3))}{x^{2} (4 x - 3)}$

Step 7.8.2

Cancel the common factors.

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Step 7.8.2.1

Factor $x$ out of $x^{2} (4 x - 3)$ .

$\frac{x (2 (2 x - 3))}{x (x (4 x - 3))}$

Step 7.8.2.2

Cancel the common factor.

$\frac{x (2 (2 x - 3))}{x (x (4 x - 3))}$

Step 7.8.2.3

Rewrite the expression.

$\frac{2 (2 x - 3)}{x (4 x - 3)}$