High School

Find the derivative of the function [tex]f(x) = x^{\frac{2}{3}} - 4x^9[/tex]. What is [tex]f'(x)[/tex]?

A. [tex]\left(\frac{2}{3}\right)x^{-\frac{1}{3}} - 36x^8[/tex]
B. [tex]\left(\frac{2}{3}\right)x^{\frac{1}{3}} - 36x^8[/tex]
C. [tex]\left(\frac{2}{3}\right)x^{-\frac{1}{3}} - 36x^9[/tex]
D. [tex]\left(\frac{2}{3}\right)x^{\frac{1}{3}} - 36x^9[/tex]

Answer :

Final answer:

The derivative f'(x) of the function f(x) = x^(2/3) - 4x^9 is found using the power rule and is (2/3)x^(-1/3) - 36x^8, which corresponds to choice (a).

Explanation:

The student has asked for the derivative f'(x) of the function f(x) = x2/3 - 4x9. To find the derivative, we apply the power rule which states that if f(x) = xn, then f'(x) = nxn-1.

Following this rule, the derivative of x2/3 is (2/3)x(2/3)-1 or (2/3)x-1/3. The derivative of -4x9 is -36x8. Combining these gives us:

f'(x) = (2/3)x-1/3 - 36x8

Thus, the correct answer is (a). (2/3)x-1/3 - 36x8.

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