Deriving Partial Derivatives for Power Functions

[BobV]

Is there a derivation for ∂f(x,y)/∂x given:

f(x,y): g(x,y)^h(x,y)

e.g. sin(x)^(x+2y)

 

[HallsofIvy]

Yes, of course. Given $f(x)= g(x)^{h(x)}$ we have $ln(f(x))= h(x)ln(g(x))$, then $\frac{1}{f(x)}\frac{df}{dx}= ln(g(x))\frac{dh}{dx}+ \frac{h(x)}{g(x)}\frac{dg}{dx}$

So $$\frac{df}{dx}= g(x)^{h(x)}\left(ln(g(x))\frac{dh}{dx}+ \frac{h(x)}{g(x)}\frac{dg}{dx}\right)$$

Of course, the same is true if g and h are functions of x and y and you are taking the derivative with respect to x because you are treating y as a constant.

(This has nothing to do with differential equations.)

 

[BobV]

Thanks

Ah, I got it, I see what you did! Sometimes when puzzled in an instant with mysterious delight the answer appears. Thanks for the surprise gift - and problem solution.