Calculus Examples

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

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

By the Sum Rule, the derivative of $2 x^{\frac{3}{4}} + 4 x^{- \frac{1}{4}}$ with respect to $x$ is $\frac{d}{d x} [2 x^{\frac{3}{4}}] + \frac{d}{d x} [4 x^{- \frac{1}{4}}]$ .

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

Step 3

Evaluate $\frac{d}{d x} [2 x^{\frac{3}{4}}]$ .

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{3}{4}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{3}{4}}]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $4$ .

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

Step 3.6.2

Subtract $4$ from $3$ .

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

Step 3.7

Move the negative in front of the fraction.

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Multiply $3$ by $2$ .

$\frac{6 x^{- \frac{1}{4}}}{4} + \frac{d}{d x} [4 x^{- \frac{1}{4}}]$

Step 3.11

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

$\frac{6}{4 x^{\frac{1}{4}}} + \frac{d}{d x} [4 x^{- \frac{1}{4}}]$

Step 3.12

Factor $2$ out of $6$ .

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

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $2$ out of $4 x^{\frac{1}{4}}$ .

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

Step 3.13.2

Cancel the common factor.

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

Step 3.13.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $4$ .

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

Step 4.6.2

Subtract $4$ from $- 1$ .

$\frac{3}{2 x^{\frac{1}{4}}} + 4 (- \frac{1}{4} x^{\frac{- 5}{4}})$

Step 4.7

Move the negative in front of the fraction.

$\frac{3}{2 x^{\frac{1}{4}}} + 4 (- \frac{1}{4} x^{- \frac{5}{4}})$

Step 4.8

Combine $x^{- \frac{5}{4}}$ and $\frac{1}{4}$ .

$\frac{3}{2 x^{\frac{1}{4}}} + 4 (- \frac{x^{- \frac{5}{4}}}{4})$

Step 4.9

Multiply $- 1$ by $4$ .

$\frac{3}{2 x^{\frac{1}{4}}} - 4 \frac{x^{- \frac{5}{4}}}{4}$

Step 4.10

Combine $- 4$ and $\frac{x^{- \frac{5}{4}}}{4}$ .

$\frac{3}{2 x^{\frac{1}{4}}} + \frac{- 4 x^{- \frac{5}{4}}}{4}$

Step 4.11

Move $x^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{2 x^{\frac{1}{4}}} + \frac{- 4}{4 x^{\frac{5}{4}}}$

Step 4.12

Factor $4$ out of $- 4$ .

$\frac{3}{2 x^{\frac{1}{4}}} + \frac{4 \cdot - 1}{4 x^{\frac{5}{4}}}$

Step 4.13

Cancel the common factors.

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Step 4.13.1

Factor $4$ out of $4 x^{\frac{5}{4}}$ .

$\frac{3}{2 x^{\frac{1}{4}}} + \frac{4 \cdot - 1}{4 (x^{\frac{5}{4}})}$

Step 4.13.2

Cancel the common factor.

$\frac{3}{2 x^{\frac{1}{4}}} + \frac{4 \cdot - 1}{4 x^{\frac{5}{4}}}$

Step 4.13.3

Rewrite the expression.

$\frac{3}{2 x^{\frac{1}{4}}} + \frac{- 1}{x^{\frac{5}{4}}}$

Step 4.14

Move the negative in front of the fraction.

$\frac{3}{2 x^{\frac{1}{4}}} - \frac{1}{x^{\frac{5}{4}}}$