differentiate ( h(x)=3 x^{5 / 3}-6 x^{1 / 6}+3 x )
differentiate ( h(x)=3 x^{5 / 3}-6 x^{1 / 6}+3 x )
Answer
Explanation:
Step1: Apply the power rule
The power rule is ((x^n)^\prime = nx^{n - 1}). For (y = 3x^{5/3}), using the power rule: ((3x^{5/3})^\prime=3\times\frac{5}{3}x^{\frac{5}{3}-1}=5x^{2/3}). For (y=-6x^{1/6}), using the power rule: ((-6x^{1/6})^\prime=-6\times\frac{1}{6}x^{\frac{1}{6}-1}=-x^{-5/6}). For (y = 3x), using the power rule: ((3x)^\prime=3\times1x^{1 - 1}=3).
Step2: Combine the derivatives
By the sum - rule of differentiation ((u + v+w)^\prime=u^\prime + v^\prime+w^\prime), where (u = 3x^{5/3}), (v=-6x^{1/6}), (w = 3x). (h^\prime(x)=(3x^{5/3})^\prime+(-6x^{1/6})^\prime+(3x)^\prime) (h^\prime(x)=5x^{2/3}-x^{-5/6}+3)
Answer:
(h^\prime(x)=5x^{2/3}-x^{-5/6}+3) (the second option)