Calculus Examples

$f (x) = \log (x^{2} - \frac{1}{x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = x^{2} - \frac{1}{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} - \frac{1}{x}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $x^{2} - \frac{1}{x}$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{1}{x}]$ .

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

Step 2.2

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

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

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

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

Step 4

Differentiate.

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Step 4.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 4.2

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

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

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

Multiply $x^{- 2}$ by $1$ .

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

Step 4.4

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

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

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $\frac{1}{x}$ by $0$ .

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

Step 4.5.2

Add $2 x + x^{- 2}$ and $0$ .

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

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

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

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

Step 5.4

Reorder the factors of $\frac{1}{x^{2} \ln (10) - \frac{\ln (10)}{x}} (2 x + \frac{1}{x^{2}})$ .

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

Step 5.5

Simplify the denominator.

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Step 5.5.1

To write $x^{2} \ln (10)$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.5.2

Combine the numerators over the common denominator.

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

Step 5.5.3

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

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Step 5.5.3.1

Move $x$ .

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

Step 5.5.3.2

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

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Step 5.5.3.2.1

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

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

Step 5.5.3.2.2

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

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

Step 5.5.3.3

Add $1$ and $2$ .

$(2 x + \frac{1}{x^{2}}) \frac{1}{\frac{x^{3} \ln (10) - \ln (10)}{x}}$

Step 5.6

Multiply the numerator by the reciprocal of the denominator.

$(2 x + \frac{1}{x^{2}}) (1 \frac{x}{x^{3} \ln (10) - \ln (10)})$

Step 5.7

Multiply $\frac{x}{x^{3} \ln (10) - \ln (10)}$ by $1$ .

$(2 x + \frac{1}{x^{2}}) \frac{x}{x^{3} \ln (10) - \ln (10)}$

Step 5.8

Multiply $2 x + \frac{1}{x^{2}}$ by $\frac{x}{x^{3} \ln (10) - \ln (10)}$ .

$\frac{(2 x + \frac{1}{x^{2}}) x}{x^{3} \ln (10) - \ln (10)}$

Step 5.9

Simplify the numerator.

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Step 5.9.1

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{(\frac{2 x \cdot x^{2}}{x^{2}} + \frac{1}{x^{2}}) x}{x^{3} \ln (10) - \ln (10)}$

Step 5.9.2

Combine the numerators over the common denominator.

$\frac{\frac{2 x \cdot x^{2} + 1}{x^{2}} x}{x^{3} \ln (10) - \ln (10)}$

Step 5.9.3

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

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Step 5.9.3.1

Move $x^{2}$ .

$\frac{\frac{2 (x^{2} x) + 1}{x^{2}} x}{x^{3} \ln (10) - \ln (10)}$

Step 5.9.3.2

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

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Step 5.9.3.2.1

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

$\frac{\frac{2 (x^{2} x^{1}) + 1}{x^{2}} x}{x^{3} \ln (10) - \ln (10)}$

Step 5.9.3.2.2

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

$\frac{\frac{2 x^{2 + 1} + 1}{x^{2}} x}{x^{3} \ln (10) - \ln (10)}$

Step 5.9.3.3

Add $2$ and $1$ .

$\frac{\frac{2 x^{3} + 1}{x^{2}} x}{x^{3} \ln (10) - \ln (10)}$

Step 5.10

Combine $\frac{2 x^{3} + 1}{x^{2}}$ and $x$ .

$\frac{\frac{(2 x^{3} + 1) x}{x^{2}}}{x^{3} \ln (10) - \ln (10)}$

Step 5.11

Reduce the expression $\frac{(2 x^{3} + 1) x}{x^{2}}$ by cancelling the common factors.

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Step 5.11.1

Factor $x$ out of $(2 x^{3} + 1) x$ .

$\frac{\frac{x (2 x^{3} + 1)}{x^{2}}}{x^{3} \ln (10) - \ln (10)}$

Step 5.11.2

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

$\frac{\frac{x (2 x^{3} + 1)}{x \cdot x}}{x^{3} \ln (10) - \ln (10)}$

Step 5.11.3

Cancel the common factor.

$\frac{\frac{x (2 x^{3} + 1)}{x \cdot x}}{x^{3} \ln (10) - \ln (10)}$

Step 5.11.4

Rewrite the expression.

$\frac{\frac{2 x^{3} + 1}{x}}{x^{3} \ln (10) - \ln (10)}$

Step 5.12

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{3} + 1}{x} \cdot \frac{1}{x^{3} \ln (10) - \ln (10)}$

Step 5.13

Multiply $\frac{2 x^{3} + 1}{x}$ by $\frac{1}{x^{3} \ln (10) - \ln (10)}$ .

$\frac{2 x^{3} + 1}{x (x^{3} \ln (10) - \ln (10))}$