Could anybody please translate this exercise into English? A friend of mine sent me the translation via Facebook but I still don't understand why I have to first do the logaritmization of both sides of the equation and then the differentiation...
Another thing - the last sentence.... They say that it's an implicit function? It does not look like an implicit function to me.... I'm just confused because of the Spanish language.... Maybe they push me to logaritmize the whole equation so that it becomes an implicit function...
Logarithmic differentiation: A procedure to compute $\frac{dy}{dx}$ with $y=f(x)^{g(x)}$, is to apply first the logarithmic function to both sides of the equation $y=f(x)^{g(x)}$, that is, $$\ln(y)=\ln(f(x)^{g(x)})\ \Longleftrightarrow\ \ln(y)=g(x)\ln(f(x))$$ and then derive implicitly to find $\frac{dy}{dx}$. Apply this procedure to find $\frac{dy}{dx}$ with $y=x^{\sqrt{x^3+\sin(x)}}$.
This is the translation. The reason to do so is simply that it is easy to derivate the product of two functions while the original function is more complicated to derive directly.