QuestionJuly 14, 2025

Calculate f'(x) given f(x)=(3x^2-6x)^(5)/(4) f'(x)=(5)/(4)(6x-6)^1/4 f'(x)=(6x-6)^5/4 f'(x)=(5)/(4)(3x^2-6x)^1/4cdot (6x-6) f'(x)=(5)/(4)(3x^2-6x)^1/4

Calculate f'(x) given f(x)=(3x^2-6x)^(5)/(4) f'(x)=(5)/(4)(6x-6)^1/4 f'(x)=(6x-6)^5/4 f'(x)=(5)/(4)(3x^2-6x)^1/4cdot (6x-6) f'(x)=(5)/(4)(3x^2-6x)^1/4
Calculate f'(x) given
f(x)=(3x^2-6x)^(5)/(4)
f'(x)=(5)/(4)(6x-6)^1/4
f'(x)=(6x-6)^5/4
f'(x)=(5)/(4)(3x^2-6x)^1/4cdot (6x-6)
f'(x)=(5)/(4)(3x^2-6x)^1/4

Solution
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Answer

f'(x) = \frac{15}{2}(3x^2 - 6x)^{\frac{1}{4}}(x - 1) Explanation 1. Apply the Chain Rule Use the chain rule: If f(x) = (u(x))^n, then f'(x) = n(u(x))^{n-1} \cdot u'(x). 2. Identify Inner Function and Derivative Let u(x) = 3x^2 - 6x. Then, u'(x) = 6x - 6. 3. Differentiate Using the Chain Rule f'(x) = \frac{5}{4}(3x^2 - 6x)^{\frac{1}{4}} \cdot (6x - 6).

Explanation

1. Apply the Chain Rule<br /> Use the chain rule: If $f(x) = (u(x))^n$, then $f'(x) = n(u(x))^{n-1} \cdot u'(x)$.<br />2. Identify Inner Function and Derivative<br /> Let $u(x) = 3x^2 - 6x$. Then, $u'(x) = 6x - 6$.<br />3. Differentiate Using the Chain Rule<br /> $f'(x) = \frac{5}{4}(3x^2 - 6x)^{\frac{1}{4}} \cdot (6x - 6)$.
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