Q. 17 - Calculus

Question

In example (2b) we calculated a derivative two different ways use algebra to simplify the first answer and show that it is equal to the second

Step-by-Step Solution

Answer

We simplify the first part andd prove that it is equal to derivative calculated by using second method

Step 1: Given information

We are given 

$\frac{d}{dx}\left(g\right(x\left)\right)=\frac{\left(2x\right)\left(x^{-3}\sqrt{x}\right)-\left(x^2\right)(-3x^{-4}\sqrt{x}+x^{-3}({\displaystyle \frac{1}{2}}{x}^{-{\displaystyle \frac{1}{2}}}\left)\right)}{{\left(x^{-3}\sqrt{x}\right)}^2}$

Step 2: Simplify

We get,

$$\begin{gathered} \frac{d}{dx}\left(g\right(x\left)\right)=\frac{2x^{1-3+{\displaystyle \frac{1}{2}}}+3x^{2-4+{\displaystyle \frac{1}{2}}}-{\displaystyle \frac{1}{2}}{x}^{2-3-{\displaystyle \frac{1}{2}}}}{{\left(x^{-3+{\displaystyle \frac{1}{2}}}\right)}^2} \\ \frac{d}{dx}\left(g\right(x\left)\right)=\frac{2x^{-{\displaystyle \frac{3}{2}}}+3x^{-{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{1}{2}}{x}^{-{\displaystyle \frac{3}{2}}}}{x^{-5}} \\ \frac{d}{dx}\left(g\right(x\left)\right)=\frac{9}{2}{x}^{-\frac{3}{2}+5} \\ \frac{d}{dx}\left(g\right(x\left)\right)=\frac{9}{2}{x}^{\frac{7}{2}} \end{gathered}$$