Question

differentiate ( h(x)=3 x^{5 / 3}-6 x^{1 / 6}+3 x )

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)