Question

How do you differentiate `f(x)=x/cotsqrtx` using the chain rule?

Answer 1

`color(red)(f' (x)=(2sqrtx*cot sqrt(x)+xcsc^2 sqrt(x))/(2sqrtx*cot^2 sqrt(x)))`

Explanation:

From the given `f(x)=x/(cot sqrt(x))`

Use the Quotient Formula for finding derivatives

`d/dx(u/v)=(v*d/dx(u)-u*d/dx(v))/v^2`

Let `u=x` and `v=cot sqrt(x)`

`f' (x)=d/dx(x/cot sqrt(x))=(cot sqrt(x)*d/dx(x)-x*d/dx(cot sqrt(x)))/(cot sqrt(x))^2`

`f' (x)=(cot sqrt(x)+xcsc^2 sqrt(x)*1/(2sqrt(x)))/(cot sqrt(x))^2`

`color(red)(f' (x)=(2sqrtx*cot sqrt(x)+xcsc^2 sqrt(x))/(2sqrtx*cot^2 sqrt(x)))`

God bless....I hope the explanation is useful.