Find dy/dx y = fifth root of 3x

$y = \sqrt[5]{3 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3 x}$ as ${(3 x)}^{\frac{1}{5}}$ .

$y = {(3 x)}^{\frac{1}{5}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(3 x)}^{\frac{1}{5}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

Simplify with factoring out.

Factor $3$ out of $3 x$ .

$\frac{d}{d x} [{(3 (x))}^{\frac{1}{5}}]$

Apply the product rule to $3 (x)$ .

$\frac{d}{d x} [3^{\frac{1}{5}} x^{\frac{1}{5}}]$

Since $3^{\frac{1}{5}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{5}} x^{\frac{1}{5}}$ with respect to $x$ is $3^{\frac{1}{5}} \frac{d}{d x} [x^{\frac{1}{5}}]$ .

$3^{\frac{1}{5}} \frac{d}{d x} [x^{\frac{1}{5}}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{5}$ .

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Combine $- 1$ and $\frac{5}{5}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

Combine $3^{\frac{1}{5}}$ and $\frac{x^{- \frac{4}{5}}}{5}$ .

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

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

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{3^{\frac{1}{5}}}{5 x^{\frac{4}{5}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3^{\frac{1}{5}}}{5 x^{\frac{4}{5}}}$

Find dy/dx y = fifth root of 3x

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