Question

If `f(x)= cot5 x` and `g(x) = sqrt(-x+3`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

`(df)/(dx)=-(csc^2(sqrt(-x+3)))/(2sqrt(-x+3))`

Explanation:

As `f(x)=cot5x` and `g(x)=sqrt(-x+3)`, therefore

`f(g(x))=cot(5sqrt(-x+3))`

According to chain rule when `f=f(g(x))`,

`(df)/(dx)=(df)/(dg)xx(dg)/(dx)`

As `f=cot(g(x))`,

`(df)/(dg)=-csc^2(g(x)=-csc^2(sqrt(-x+3))`

and as `g(x)=sqrt(-x+3)`

`(dg)/(dx)=1/(2sqrt(-x+3))xxd/(dx)(-x+3)=-1/(2sqrt(-x+3))`

(note that we have again applied chain rule here as `-x+3` is again a function of `x` and hence we should multiply by `d/(dx)(-x+3)`)

Hence `(df)/(dx)=-csc^2(sqrt(-x+3))xx-1/(2sqrt(-x+3))`

= `-(csc^2(sqrt(-x+3)))/(2sqrt(-x+3))`