Say, I have an equation $y^2=x^3$. So I can say $dy^2/dx=3x^2$. So the very small increment in $y^2$ when $x$ becomes $dx$ is $dy^2$ which in this case is $3x^2dx$. I also know that $dy^2/dy=2y$ so I can say $dy^2=2ydy$ and equate the increments but how do I know the increments $dy^2$ are same in both the cases?
Asked
Active
Viewed 26 times
1 Answers
2
Divide the second equation you obtained by $dx$ on both sides $$\frac{dy^2}{dx} = 2y \frac{dy}{dx}$$ Then substitute the values $y = x^{\frac 32}$ and $\frac{dy}{dx} = \frac{3}{2}\sqrt{x}$, which are obtained from the initial equation $y^2 = x^3$. This gives you $$\frac{d^2y}{dx} = 3x^2$$ which is identical to equation 1.
Aniruddha Deb
- 4,345
- 11
- 30
-
but how do I know the increments $dy^2$ are same in both cases - I answered this question. – Aniruddha Deb Jun 07 '20 at 07:08