Calculus Examples

$f (x) = x - \frac{4}{\sqrt[4]{x^{3}}}$

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate $\frac{d}{d x} [- \frac{4}{\sqrt[4]{x^{3}}}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

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

Step 2.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4}{x^{\frac{3}{4}}}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{1}{x^{\frac{3}{4}}}]$ .

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

Step 2.3

Rewrite $\frac{1}{x^{\frac{3}{4}}}$ as ${(x^{\frac{3}{4}})}^{- 1}$ .

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

Step 2.4

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 $g (x) = x^{\frac{3}{4}}$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{4}$ .

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

Step 2.6

Multiply the exponents in ${(x^{\frac{3}{4}})}^{- 2}$ .

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Step 2.6.1

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

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

Step 2.6.2

Cancel the common factor of $2$ .

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Step 2.6.2.1

Factor $2$ out of $4$ .

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

Step 2.6.2.2

Factor $2$ out of $- 2$ .

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

Step 2.6.2.3

Cancel the common factor.

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

Step 2.6.2.4

Rewrite the expression.

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

Step 2.6.3

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 2.6.4

Multiply $3$ by $- 1$ .

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

Step 2.6.5

Move the negative in front of the fraction.

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

Step 2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.8

Combine $- 1$ and $\frac{4}{4}$ .

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $4$ .

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

Step 2.10.2

Subtract $4$ from $3$ .

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

Step 2.11

Move the negative in front of the fraction.

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

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

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Step 2.14.1

Move $x^{- \frac{3}{2}}$ .

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

Step 2.14.2

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

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

Step 2.14.3

To write $- \frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.14.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.14.4.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

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

Step 2.14.4.2

Multiply $2$ by $2$ .

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

Step 2.14.5

Combine the numerators over the common denominator.

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

Step 2.14.6

Simplify the numerator.

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Step 2.14.6.1

Multiply $- 3$ by $2$ .

$1 - 4 (- \frac{3 x^{\frac{- 6 - 1}{4}}}{4})$

Step 2.14.6.2

Subtract $1$ from $- 6$ .

$1 - 4 (- \frac{3 x^{\frac{- 7}{4}}}{4})$

Step 2.14.7

Move the negative in front of the fraction.

$1 - 4 (- \frac{3 x^{- \frac{7}{4}}}{4})$

Step 2.15

Move $x^{- \frac{7}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$1 - 4 (- \frac{3}{4 x^{\frac{7}{4}}})$

Step 2.16

Multiply $- 1$ by $- 4$ .

$1 + 4 \frac{3}{4 x^{\frac{7}{4}}}$

Step 2.17

Combine $4$ and $\frac{3}{4 x^{\frac{7}{4}}}$ .

$1 + \frac{4 \cdot 3}{4 x^{\frac{7}{4}}}$

Step 2.18

Multiply $4$ by $3$ .

$1 + \frac{12}{4 x^{\frac{7}{4}}}$

Step 2.19

Factor $4$ out of $12$ .

$1 + \frac{4 \cdot 3}{4 x^{\frac{7}{4}}}$

Step 2.20

Cancel the common factors.

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Step 2.20.1

Factor $4$ out of $4 x^{\frac{7}{4}}$ .

$1 + \frac{4 \cdot 3}{4 (x^{\frac{7}{4}})}$

Step 2.20.2

Cancel the common factor.

$1 + \frac{4 \cdot 3}{4 x^{\frac{7}{4}}}$

Step 2.20.3

Rewrite the expression.

$1 + \frac{3}{x^{\frac{7}{4}}}$