Question

let f(x)=sin(lnx)

a. What is the range of the function f?
b. Calculate f ’(x). Use calculus to find the maximum value of f on the interval [1,10]

Answer 1

well, what is the image (range) of \(\log x\)? well we know that \(\log x\) has an image of \((-\infty,\infty)\). what about the image of \(\sin x\)? well, we know that \(\sin x\) has an image of \([-1,1]\)... ergo the image of \(\sin(\log x)\) is just \([-1,1]\) (since the image of \(\log x\) is identical to the domain of \(\sin x\))

Answer 2

now consider \(f(x)=\sin(\log x)\). to differentiate use the chain rule:$$f'(x)=\frac{d}{dx}\sin(\log x)=\frac{d}{d(\log x)}\sin(\log x)\cdot\frac{d}{dx}\log(x)=\cos(\log x)\cdot\frac1x$$

Answer 3

so what does that mean @oldrin.bataku