Calculus Examples

$f (x) = x^{\frac{27}{13}} + x^{\frac{14}{13}}$

Step 1

By the Sum Rule, the derivative of $x^{\frac{27}{13}} + x^{\frac{14}{13}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{27}{13}}] + \frac{d}{d x} [x^{\frac{14}{13}}]$ .

$\frac{d}{d x} [x^{\frac{27}{13}}] + \frac{d}{d x} [x^{\frac{14}{13}}]$

Step 2

Evaluate $\frac{d}{d x} [x^{\frac{27}{13}}]$ .

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{27}{13}$ .

$\frac{27}{13} x^{\frac{27}{13} - 1} + \frac{d}{d x} [x^{\frac{14}{13}}]$

Step 2.2

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

$\frac{27}{13} x^{\frac{27}{13} - 1 \cdot \frac{13}{13}} + \frac{d}{d x} [x^{\frac{14}{13}}]$

Step 2.3

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

$\frac{27}{13} x^{\frac{27}{13} + \frac{- 1 \cdot 13}{13}} + \frac{d}{d x} [x^{\frac{14}{13}}]$

Step 2.4

Combine the numerators over the common denominator.

$\frac{27}{13} x^{\frac{27 - 1 \cdot 13}{13}} + \frac{d}{d x} [x^{\frac{14}{13}}]$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $13$ .

$\frac{27}{13} x^{\frac{27 - 13}{13}} + \frac{d}{d x} [x^{\frac{14}{13}}]$

Step 2.5.2

Subtract $13$ from $27$ .

$\frac{27}{13} x^{\frac{14}{13}} + \frac{d}{d x} [x^{\frac{14}{13}}]$

Step 3

Evaluate $\frac{d}{d x} [x^{\frac{14}{13}}]$ .

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{14}{13}$ .

$\frac{27}{13} x^{\frac{14}{13}} + \frac{14}{13} x^{\frac{14}{13} - 1}$

Step 3.2

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

$\frac{27}{13} x^{\frac{14}{13}} + \frac{14}{13} x^{\frac{14}{13} - 1 \cdot \frac{13}{13}}$

Step 3.3

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

$\frac{27}{13} x^{\frac{14}{13}} + \frac{14}{13} x^{\frac{14}{13} + \frac{- 1 \cdot 13}{13}}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{27}{13} x^{\frac{14}{13}} + \frac{14}{13} x^{\frac{14 - 1 \cdot 13}{13}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $13$ .

$\frac{27}{13} x^{\frac{14}{13}} + \frac{14}{13} x^{\frac{14 - 13}{13}}$

Step 3.5.2

Subtract $13$ from $14$ .

$\frac{27}{13} x^{\frac{14}{13}} + \frac{14}{13} x^{\frac{1}{13}}$

Step 4

Simplify each term.

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Step 4.1

Combine $\frac{27}{13}$ and $x^{\frac{14}{13}}$ .

$\frac{27 x^{\frac{14}{13}}}{13} + \frac{14}{13} x^{\frac{1}{13}}$

Step 4.2

Combine $\frac{14}{13}$ and $x^{\frac{1}{13}}$ .

$\frac{27 x^{\frac{14}{13}}}{13} + \frac{14 x^{\frac{1}{13}}}{13}$