I'm self-studying through some exercises on differentiation, and have found a section which I've gotten almost every question wrong. Can anyone help me find out where I'm going wrong?
An example of one of the questions and my attempt to answer it is
Q: Find $\frac{dy}{dx}$ in terms of $y$ when $x = 4y^5-8\sqrt{y}$
My answer: $x = 4y^5-8\sqrt{y} = 4y^5 - 8y^\frac{1}{2}$ $\frac{dx}{dy} = 20y^4 - 4y^{-\frac{1}{2}}$ $ = 20y^4-\frac{4}{\sqrt{y}}$ since: $ \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} \\ \therefore \frac{dy}{dx} = \frac{1}{20y^4}-\frac {y^{-\frac{1}{2}}}{4} \\ = \frac {4 - 20(y^{-\frac{1}{2}})}{80y^4} \\ = \frac {1 - 5(y^{-\frac{1}{2}})}{20y^4} $
But the listed answer is: $\frac{y^\frac{1}{2}}{4(5y^\frac{9}{2}-1)}$
I suspect that due to the fact I've gotten only 2 out of the 6 questions right I'm doing something fundamentally wrong! Can anyone help me figure out either where I'm going wrong, or what I'm missing?