Answer
$$\frac{dp}{dq}=-\frac{(5p^2+2p)^{5/2}}{3(5p+1)}$$
Work Step by Step
$$q=(5p^2+2p)^{-3/2}$$
To find $dp/dq$, we would use the methods of implicit differentiation.
Differentiate both sides of the equation with respect to $q$: $$1=-\frac{3}{2}(5p^2+2p)^{-5/2}(5p^2+2p)'$$
$$1=-\frac{3}{2}(5p^2+2p)^{-5/2}\Big(10p\frac{dp}{dq}+2\frac{dp}{dq}\Big)$$
$$1=-\frac{3}{2}(5p^2+2p)^{-5/2}(5p+1)2\frac{dp}{dq}$$
$$1=-3(5p^2+2p)^{-5/2}(5p+1)\frac{dp}{dq}$$
Then calculate for $dp/dq$: $$\frac{dp}{dq}=-\frac{1}{3(5p^2+2p)^{-5/2}(5p+1)}$$
$$\frac{dp}{dq}=-\frac{(5p^2+2p)^{5/2}}{3(5p+1)}$$
