Calculus Examples

$f (x) = 8 x + 13 x^{\frac{8}{13}}$

Step 1

By the Sum Rule, the derivative of $8 x + 13 x^{\frac{8}{13}}$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [13 x^{\frac{8}{13}}]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [13 x^{\frac{8}{13}}]$

Step 2

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x] + \frac{d}{d x} [13 x^{\frac{8}{13}}]$

Step 2.2

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

$8 \cdot 1 + \frac{d}{d x} [13 x^{\frac{8}{13}}]$

Step 2.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [13 x^{\frac{8}{13}}]$

Step 3

Evaluate $\frac{d}{d x} [13 x^{\frac{8}{13}}]$ .

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Step 3.1

Since $13$ is constant with respect to $x$ , the derivative of $13 x^{\frac{8}{13}}$ with respect to $x$ is $13 \frac{d}{d x} [x^{\frac{8}{13}}]$ .

$8 + 13 \frac{d}{d x} [x^{\frac{8}{13}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{8}{13}$ .

$8 + 13 (\frac{8}{13} x^{\frac{8}{13} - 1})$

Step 3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$8 + 13 (\frac{8}{13} x^{\frac{8}{13} - 1 \cdot \frac{13}{13}})$

Step 3.4

Combine $- 1$ and $\frac{13}{13}$ .

$8 + 13 (\frac{8}{13} x^{\frac{8}{13} + \frac{- 1 \cdot 13}{13}})$

Step 3.5

Combine the numerators over the common denominator.

$8 + 13 (\frac{8}{13} x^{\frac{8 - 1 \cdot 13}{13}})$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $13$ .

$8 + 13 (\frac{8}{13} x^{\frac{8 - 13}{13}})$

Step 3.6.2

Subtract $13$ from $8$ .

$8 + 13 (\frac{8}{13} x^{\frac{- 5}{13}})$

Step 3.7

Move the negative in front of the fraction.

$8 + 13 (\frac{8}{13} x^{- \frac{5}{13}})$

Step 3.8

Combine $\frac{8}{13}$ and $x^{- \frac{5}{13}}$ .

$8 + 13 \frac{8 x^{- \frac{5}{13}}}{13}$

Step 3.9

Combine $13$ and $\frac{8 x^{- \frac{5}{13}}}{13}$ .

$8 + \frac{13 (8 x^{- \frac{5}{13}})}{13}$

Step 3.10

Multiply $8$ by $13$ .

$8 + \frac{104 x^{- \frac{5}{13}}}{13}$

Step 3.11

Move $x^{- \frac{5}{13}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 + \frac{104}{13 x^{\frac{5}{13}}}$

Step 3.12

Factor $13$ out of $104$ .

$8 + \frac{13 \cdot 8}{13 x^{\frac{5}{13}}}$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $13$ out of $13 x^{\frac{5}{13}}$ .

$8 + \frac{13 \cdot 8}{13 (x^{\frac{5}{13}})}$

Step 3.13.2

Cancel the common factor.

$8 + \frac{13 \cdot 8}{13 x^{\frac{5}{13}}}$

Step 3.13.3

Rewrite the expression.

$8 + \frac{8}{x^{\frac{5}{13}}}$