I'm having a bit of trouble finding the second implicit derivative for the equation $y^5 = x^8$.
I have the first derivative, which is $8x^7 / 5y^4$, but I'm having trouble after that. I did the quotient rule and got $280x^6y^4 – 160x^7y^3 / (20y^3)^2$. I'm not quite sure what to do after this. Any help would be appreciated. Thanks!
Best Answer
In your quotient rule differentiation, there was an error. The derivative of $5y^4$ with respect to $x$ should be $20y^3\frac{dy}{dx}$. Then substitute the value of $\frac{dy}{dx}$ that you found in your first calculation, and, possibly, simplify.
I prefer to do things a little differently. After the first differentiation, we have $$5y^4\frac{dy}{dx}=8x^7.$$ Now differentiate again. We get $$5y^4\frac{d^2 y}{dx^2}+20y^3\left(\frac{dy}{dx}\right)^2=56x^6.$$ Substitute the value you found for $\frac{dy}{dx}$, and solve for $\frac{d^2 y}{dx^2}$.