Here's a cute puzzle:

If you have the function $\displaystyle h(x)= f(x)^{g(x)}$, where f and g are differentiable functions, then there twomistakesone could make in trying to find the derivative of h.

1) Treat g(x) as if it were constant and use the "power rule" (very common): $\displaystyle (f^g)'= g(x)f(x)^{g(x)- 1}$.

2) tread f(x) as if it were constant and use the "exponential rule" (much less common): $\displaystyle (f^g)'= ln(f(x))(f(x)^{g(x)})g'(x)$.

Those are, as I said,wrong. Show that the correct derivative is thesumof those two mistakes!