Find the Derivative - d/d@VAR f(x)=2x-6x^(1/3)

$f (x) = 2 x - 6 x^{\frac{1}{3}}$

Step 1

By the Sum Rule, the derivative of $2 x - 6 x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6 x^{\frac{1}{3}}]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 6 x^{\frac{1}{3}}]$

Step 2

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

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 6 x^{\frac{1}{3}}]$

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

$2 \cdot 1 + \frac{d}{d x} [- 6 x^{\frac{1}{3}}]$

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 6 x^{\frac{1}{3}}]$

Step 3

Evaluate $\frac{d}{d x} [- 6 x^{\frac{1}{3}}]$ .

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{\frac{1}{3}}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$2 - 6 \frac{d}{d x} [x^{\frac{1}{3}}]$

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

$2 - 6 (\frac{1}{3} x^{\frac{1}{3} - 1})$

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

$2 - 6 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

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

$2 - 6 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Combine the numerators over the common denominator.

$2 - 6 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})$

Simplify the numerator.

Multiply $- 1$ by $3$ .

$2 - 6 (\frac{1}{3} x^{\frac{1 - 3}{3}})$

Subtract $3$ from $1$ .

$2 - 6 (\frac{1}{3} x^{\frac{- 2}{3}})$

Move the negative in front of the fraction.

$2 - 6 (\frac{1}{3} x^{- \frac{2}{3}})$

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

$2 - 6 \frac{x^{- \frac{2}{3}}}{3}$

Combine $- 6$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

$2 + \frac{- 6 x^{- \frac{2}{3}}}{3}$

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 + \frac{- 6}{3 x^{\frac{2}{3}}}$

Factor $3$ out of $- 6$ .

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

Cancel the common factors.

Factor $3$ out of $3 x^{\frac{2}{3}}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Find the Derivative - d/d@VAR f(x)=2x-6x^(1/3)

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