Calculus Examples

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

Step 1

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

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

Step 2

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

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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 $n = \frac{1}{3}$ .

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

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $3$ .

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

Step 2.5.2

Subtract $3$ from $1$ .

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

Step 2.6

Move the negative in front of the fraction.

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

Step 3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{3} x^{- \frac{2}{3}} + 0$

Step 4

Simplify.

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Step 4.1

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

$\frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}} + 0$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

$\frac{1}{3 x^{\frac{2}{3}}} + 0$

Step 4.2.2

Add $\frac{1}{3 x^{\frac{2}{3}}}$ and $0$ .

$\frac{1}{3 x^{\frac{2}{3}}}$