Let $p, q\geq 2$, $s\geq p$ and $f,g$ be non-negative smooth enough functions. Then why does the following inequality hold: $$-f^{q-2}g^{s}|\nabla f|^{p}+f^{q-1}g^{s-1}|\nabla f|^{p-1}|\nabla g|\leq C(s, q)(-|\nabla (f^{\frac{p+q-2}{p}})|^{p}g^{s}+|\nabla g|^{p}g^{s-p}f^{p+q-2}),$$ for some constant depending $C(s, q)$ on $q$ and $s$?

In the source I have, it is said that this follows from Young's inequality, but I do not know with which exponents and applied to which function.

Any help is appreciated!

Let $p, q\geq 2$, $s\geq p$ and $f,g$ be non-negative smooth enough functions. Then why does the following inequality hold: $$-f^{q-2}g^{s}|\nabla f|^{p}+f^{q-1}g^{s-1}|\nabla f|^{p-1}|\nabla g|\leq C(s, q)(-|\nabla (f^{\frac{p+q-2}{p}})|^{p}g^{s}+|\nabla g|^{p}g^{s-p}f^{p+q-2}),$$ for some constant depending $C(s, q)$ on $q$ and $s$?

In the source [1] (Proof of Lemma 3.1) I have, it is said that this follows from Young's inequality, but I do not know with which exponents and applied to which function.

Any help is appreciated!

[1] (S.A.J. Dekkers "Finite propagation speed for solutions of the parabolic p-Laplace equation on manifolds" Comm. in Analysis and Geometry, 13 (2005), no.4, 741-768)