Calculus Examples

$f (x) = \frac{1}{x} + \frac{1}{x^{3}} - \frac{4}{\sqrt{x}}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{x} + \frac{1}{x^{3}} - \frac{4}{\sqrt{x}}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [- \frac{4}{\sqrt{x}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{x^{3}}]$ .

Tap for more steps...

Step 3.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

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

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3.4

Multiply the exponents in ${(x^{3})}^{- 2}$ .

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Multiply $3$ by $- 2$ .

$- x^{- 2} - x^{- 6} (3 x^{2}) + \frac{d}{d x} [- \frac{4}{\sqrt{x}}]$

Step 3.5

Multiply $3$ by $- 1$ .

$- x^{- 2} - 3 x^{- 6} x^{2} + \frac{d}{d x} [- \frac{4}{\sqrt{x}}]$

Step 3.6

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.6.1

Move $x^{2}$ .

$- x^{- 2} - 3 (x^{2} x^{- 6}) + \frac{d}{d x} [- \frac{4}{\sqrt{x}}]$

Step 3.6.2

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

$- x^{- 2} - 3 x^{2 - 6} + \frac{d}{d x} [- \frac{4}{\sqrt{x}}]$

Step 3.6.3

Subtract $6$ from $2$ .

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

Step 4

Evaluate $\frac{d}{d x} [- \frac{4}{\sqrt{x}}]$ .

Tap for more steps...

Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4}{x^{\frac{1}{2}}}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

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

Step 4.3

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

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

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.5

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

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

Step 4.6

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

Tap for more steps...

Step 4.6.1

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

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

Step 4.6.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.6.2.1

Factor $2$ out of $- 2$ .

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

Step 4.6.2.2

Cancel the common factor.

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

Step 4.6.2.3

Rewrite the expression.

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

Step 4.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.8

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 4.9

Combine the numerators over the common denominator.

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

Step 4.10

Simplify the numerator.

Tap for more steps...

Step 4.10.1

Multiply $- 1$ by $2$ .

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

Step 4.10.2

Subtract $2$ from $1$ .

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

Step 4.11

Move the negative in front of the fraction.

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

Step 4.12

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

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

Step 4.13

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

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

Step 4.14

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

Tap for more steps...

Step 4.14.1

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

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

Step 4.14.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.14.3

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 4.14.4

Combine the numerators over the common denominator.

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

Step 4.14.5

Simplify the numerator.

Tap for more steps...

Step 4.14.5.1

Multiply $- 1$ by $2$ .

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

Step 4.14.5.2

Subtract $2$ from $- 1$ .

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

Step 4.14.6

Move the negative in front of the fraction.

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

Step 4.15

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.16

Multiply $- 1$ by $- 4$ .

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

Step 4.17

Combine $4$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

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

Step 4.18

Factor $2$ out of $4$ .

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

Step 4.19

Cancel the common factors.

Tap for more steps...

Step 4.19.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

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

Step 4.19.2

Cancel the common factor.

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

Step 4.19.3

Rewrite the expression.

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

Step 5

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

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

Step 6

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

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

Step 7

Combine terms.

Tap for more steps...

Step 7.1

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

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

Step 7.2

Move the negative in front of the fraction.

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