differentiation

how would you differentiate

y=sin(In x)

what rule would you use and why?

Chain because you have a function inside a function.

Bonus points if you spot what function is inside which.

thank you .
ln x is inside sin ? idk lol

If you don't mind , could you tell me when to use the 3 rules because the responses i get from this forum are totally different from my teacher .

Exactly.



The names are self-explanatory.


You use the product rule when you wish to differentiate an expression that is a PRODUCT of two functions in the form $f(x)g(x)$, e.g. $x^2 \sin(x)$, or $e^x \log x$

You use the quotient rule when you wish to differentiate an expression that is a QUOTIENT of two functions in the form $\dfrac{f(x)}{g(x)}$, e.g. $\dfrac{e^x}{\sin(x)}$, or $\dfrac{x^3+1}{x^5+x^3+x+1}$

You use the chain rule when you wish to differentiate an expression that is a COMPOSITION of two functions in the form $f(g(x))$, e.g. $\sin(x^2)$, or $e^{x^3}$


Is this not what your teacher said...?

P.S. The quotient rule can often be avoided by simply rewriting $\dfrac{f(x)}{g(x)}$ as $f(x)g(x)^{-1}$ and then using the product rule + chain rule. I.e. $\dfrac{x^2}{e^x} = x^2e^{-x}$