Q 91 - Calculus

Question

Use implicit differentiation and the fact that $\frac{d}{dx}\left(x^3\right)=3x^2$ and $\frac{d}{dx}\left(x^5\right)=5x^4$ to prove that $\frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^{-2/5}$. 

Step-by-Step Solution

Answer

$$\begin{gathered} y=x^{3/5} \\ \Rightarrow y^5=x^3 \\ \Rightarrow 5y^4\frac{dy}{dx}=3x^2 \\ \Rightarrow \frac{dy}{dx}=\frac{3x^2}{5y^4} \\ \Rightarrow \frac{dy}{dx}=\frac{3}{5}{x}^2{y}^{-4} \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^2{\left(x^{3/5}\right)}^{-4} \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^2\left(x^{3/5\times -4}\right) \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^2\left(x^{-12/5}\right) \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^{2-12/5} \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^{-2/5} \end{gathered}$$

Hence proved.

Step 1. Given Information

We have given that :-

$\frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^{-2/5}$.

We have to prove this derivative by using implicit differentiation and the following derivatives :- 

$\frac{d}{dx}\left(x^3\right)=3x^2$ and $\frac{d}{dx}\left(x^5\right)=5x^4$.

Step 2. Prove that d d x ( x 3 / 5 ) = 3 5 x - 2 / 5

We have to find the derivative of $x^{3/5}$.

Let :-

$y=x^{3/5}$.

We can write it as :- 

$y^5=x^3$.

Then by using implicit differentiation, given values $\frac{d}{dx}\left(x^3\right)=3x^2$and $\frac{d}{dx}\left(x^5\right)=5x^4$ and chain rule, we have :-

$$\begin{gathered} 5y^4\frac{dy}{dx}=3x^2 \\ \Rightarrow \frac{dy}{dx}=\frac{3x^2}{5y^4} \\ \Rightarrow \frac{dy}{dx}=\frac{3}{5}{x}^2{y}^{-4} \end{gathered}$$

Put the value $y=x^{3/5}$, then we have :-

$$\begin{gathered} \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^2{\left(x^{3/5}\right)}^{-4} \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^2\left(x^{3/5\times -4}\right) \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^2\left(x^{-12/5}\right) \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^{2-12/5} \\ \Rightarrow \frac{d}{dx}\left(x^{3/5}\right)=\frac{3}{5}{x}^{-2/5} \end{gathered}$$

This is the required value.

Hence proved.