Differentiate y=ln(ln(x)) with respect to x.
To solve this question we need to understand the process of implicit differentiation, which is a case of using the chain rule. If you remember the chain rule states that for y=f(g(x)), we have y'=f'(g(x))g'(x), so that we treat y as being composed of two functions and differentiate them individually, then multiply. So instead if we have f(y)=g(x), then using the same rule on the left hand side but with y, and differentiating both sides we get y'f'(y)=g'(x). Now that this is understood we can solve the question. We are given y=ln(ln(x)) so ey = ln(x). Now differentiate this on both sides: y'ey=1/x. Now we're looking for y' on one side and everything else on the other:y'=1/(xey). We're almost there but there's a problem, we want y' with respect to x so we need the right hand side only with x: fortunately we know ey=lnx, so y'=1/(xln(x)). And we are done. Can you differentiate y = ln(ln(x2)) for me?
