Questions: Find the derivative of f(x) = √[5]x^3. a) f(x) = 3 / (5x^(2/5)) b) f(x) = 1 / (5(x^3)^2) c) f(x) = (3/5) x^(2/5) d) f(x) = (5/3) x^(2/3)

Transcript text: Find the derivative of $f(x)=\sqrt[5]{x^{3}}$. a) $f(x)=\frac{3}{5 x^{\frac{2}{5}}}$ b) $f(x)=\frac{1}{5\left(x^{3}\right)^{2}}$ c) $f(x)=\frac{3}{5} x^{\frac{2}{5}}$ d) $f(x)=\frac{5}{3} x^{\frac{2}{3}}$

Solution

Solution Steps

Step 1: Rewrite the Function

The given function is $ f(x) = \sqrt[5]{x^3} $. We can rewrite this function using exponent notation:

\[ f(x) = (x^3)^{\frac{1}{5}} = x^{\frac{3}{5}} \]

Step 2: Differentiate the Function

To find the derivative of $ f(x) = x^{\frac{3}{5}} $, we use the power rule for differentiation, which states that if $ f(x) = x^n $, then $ f'(x) = n \cdot x^{n-1} $.

Applying the power rule:

\[ f'(x) = \frac{3}{5} \cdot x^{\frac{3}{5} - 1} = \frac{3}{5} \cdot x^{-\frac{2}{5}} \]

Step 3: Simplify the Derivative

The derivative can be simplified to:

\[ f'(x) = \frac{3}{5x^{\frac{2}{5}}} \]