why does the power rule for differentiation doesn't apply to a function like y = x^(cosx)?
how do you solve y = x^(cosx)
hmm...
Because the power rule is proved for constant exponents. To differentiate something like, x^(cos(x)), simply use the inverse property of logarithms:
$\displaystyle e^{\ln{x}} = x$
$\displaystyle y = x^{\cos{x}} = e^{\ln{x^{\cos{x}}}} = e^{\cos{x}\ln{x}}$
Use chain rule to differentiate.
Another way to use the same idea is if you have $\displaystyle y=x^{\cos(x)}$, then just take the natural log of both sides and use the fact that $\displaystyle \frac{d}{dx} \ln(y) = \frac{y'}{y}$ if y is a function of x. You can say in general I think that $\displaystyle y' = y*\frac{d}{dx} \left( \ln(f[x]) \right) $ (there's some conditions on that of course). That's the way I liked doing this problems.