Question

if $y = \frac{1}{2}x^{4/5}-\frac{3}{x^{5}}$, then $\frac{dy}{dx}=$
(a) $\frac{2}{5x^{1/5}}+\frac{15}{x^{6}}$
(b) $\frac{2}{5x^{1/5}}+\frac{15}{x^{4}}$
(c) $\frac{2}{5x^{1/5}}-\frac{3}{5x^{4}}$
(d) $\frac{2x^{1/5}}{5}+\frac{15}{x^{6}}$
(e) $\frac{2x^{1/5}}{5}-\frac{3}{5x^{4}}$

Explanation:

Step1: Recall power - rule for differentiation

The power - rule states that if $y = ax^n$, then $\frac{dy}{dx}=anx^{n - 1}$. For the function $y=\frac{1}{2}x^{4/5}-\frac{3}{x^{5}}=\frac{1}{2}x^{4/5}-3x^{- 5}$.

Step2: Differentiate the first term

For $y_1=\frac{1}{2}x^{4/5}$, using the power - rule with $a = \frac{1}{2}$ and $n=\frac{4}{5}$, we have $\frac{dy_1}{dx}=\frac{1}{2}\times\frac{4}{5}x^{\frac{4}{5}-1}=\frac{2}{5}x^{-\frac{1}{5}}=\frac{2}{5x^{1/5}}$.

Step3: Differentiate the second term

For $y_2=-3x^{-5}$, using the power - rule with $a=-3$ and $n = - 5$, we get $\frac{dy_2}{dx}=-3\times(-5)x^{-5 - 1}=15x^{-6}=\frac{15}{x^{6}}$.

Step4: Find the derivative of the whole function

Since $y = y_1 + y_2$, then $\frac{dy}{dx}=\frac{dy_1}{dx}+\frac{dy_2}{dx}=\frac{2}{5x^{1/5}}+\frac{15}{x^{6}}$.

Answer:

A. $\frac{2}{5x^{1/5}}+\frac{15}{x^{6}}$