Calculus Examples

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

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 $f (x) = 1 + \ln (x)$ and $g (x) = x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.2

Multiply $2$ by $2$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

Combine $x^{2}$ and $\frac{1}{x}$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

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

Step 4.2.2.3

Cancel the common factor.

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

Step 4.2.2.4

Rewrite the expression.

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

Step 4.2.2.5

Divide $x$ by $1$ .

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

Step 4.3

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

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

Step 4.4

Simplify with factoring out.

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Step 4.4.1

Multiply $2$ by $- 1$ .

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

Step 4.4.2

Factor $x$ out of $x - 2 (1 + \ln (x)) x$ .

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Step 4.4.2.1

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

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

Step 4.4.2.2

Factor $x$ out of $x^{1}$ .

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

Step 4.4.2.3

Factor $x$ out of $- 2 (1 + \ln (x)) x$ .

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

Step 4.4.2.4

Factor $x$ out of $x \cdot 1 + x (- 2 (1 + \ln (x)))$ .

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

Step 5

Cancel the common factors.

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Step 5.1

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

$\frac{x (1 - 2 (1 + \ln (x)))}{x \cdot x^{3}}$

Step 5.2

Cancel the common factor.

$\frac{x (1 - 2 (1 + \ln (x)))}{x \cdot x^{3}}$

Step 5.3

Rewrite the expression.

$\frac{1 - 2 (1 + \ln (x))}{x^{3}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{1 - 2 \cdot 1 - 2 \ln (x)}{x^{3}}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $- 2$ by $1$ .

$\frac{1 - 2 - 2 \ln (x)}{x^{3}}$

Step 6.2.1.2

Simplify $- 2 \ln (x)$ by moving $2$ inside the logarithm.

$\frac{1 - 2 - \ln (x^{2})}{x^{3}}$

Step 6.2.2

Subtract $2$ from $1$ .

$\frac{- 1 - \ln (x^{2})}{x^{3}}$

Step 6.3

Factor $- 1$ out of $- 1 - \ln (x^{2})$ .

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Step 6.3.1

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

$\frac{- 1 (1) - \ln (x^{2})}{x^{3}}$

Step 6.3.2

Factor $- 1$ out of $- \ln (x^{2})$ .

$\frac{- 1 (1) - (\ln (x^{2}))}{x^{3}}$

Step 6.3.3

Factor $- 1$ out of $- 1 (1) - (\ln (x^{2}))$ .

$\frac{- 1 (1 + \ln (x^{2}))}{x^{3}}$

Step 6.3.4

Rewrite $- 1 (1 + \ln (x^{2}))$ as $- (1 + \ln (x^{2}))$ .

$\frac{- (1 + \ln (x^{2}))}{x^{3}}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{1 + \ln (x^{2})}{x^{3}}$