Question

In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{y=\sqrt[5]{3 x^{3}+4 x}} \quad \frac{\text { Point }}{(2,2)}\)

Short answer

The derivative of the function \(y=\sqrt[5]{3 x^{3}+4 x}\) at the point (2,2) is 0.3.

Step-by-step solution

Differentiate the function

Rewrite the function as \(y=(3 x^{3}+4 x)^{1/5}\) to ease differentiation. Use chain rule which states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function: \[y'=(1/5)*(3x^3+4x)^{-4/5}* (9x^2+4)\]

Plug in the x-coordinate of the point into the derivative

Substitute x=2 into the derivative function from Step 1 so as to get the value of the derivative at this point: \[y'(2)=(1/5)*(3*(2)^3+4*(2))^{-4/5}* (9*(2)^2+4)=0.3\]

Interpret the result

The derivative of the function \(y=\sqrt[5]{3 x^{3}+4 x}\) at the point (2,2) is 0.3. This indicates the rate of change of y with respect to x at this point.