Calculus Examples

$G (x) = {(\frac{1}{x})}^{4} - \ln (x)$

Step 1

By the Sum Rule, the derivative of ${(\frac{1}{x})}^{4} - \ln (x)$ with respect to $x$ is $\frac{d}{d x} [{(\frac{1}{x})}^{4}] + \frac{d}{d x} [- \ln (x)]$ .

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

Step 2

Evaluate $\frac{d}{d x} [{(\frac{1}{x})}^{4}]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \frac{1}{x}$ .

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Step 2.1.1

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

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

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

Step 2.1.3

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $4$ .

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

Step 3

Evaluate $\frac{d}{d x} [- \ln (x)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (x)$ with respect to $x$ is $- \frac{d}{d x} [\ln (x)]$ .

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

Step 3.2

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

$- 4 {(\frac{1}{x})}^{3} x^{- 2} - \frac{1}{x}$

Step 4

Simplify.

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Step 4.1

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

$- 4 {(\frac{1}{x})}^{3} \frac{1}{x^{2}} - \frac{1}{x}$

Step 4.2

Apply the product rule to $\frac{1}{x}$ .

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

Step 4.3

Combine terms.

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Step 4.3.1

One to any power is one.

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

Step 4.3.2

Combine $- 4$ and $\frac{1}{x^{3}}$ .

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 4.3.4

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

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

Step 4.3.5

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 4.3.5.1

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

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

Step 4.3.5.2

Add $2$ and $3$ .

$- \frac{4}{x^{5}} - \frac{1}{x}$

Step 4.4

Reorder terms.

$- \frac{1}{x} - \frac{4}{x^{5}}$