How do you differentiate #f(x)=cot(sqrt(x-3)) # using the chain rule?

1 Answer
Mar 28, 2017

Answer:

#-csc^2(sqrt(x-3))/(2*sqrt(x-3))#

Explanation:

Differentiate the outer function with respect to the inner function (ie #dy/(du)#, where du #= sqrt(x-3)# and multiply with the derivative of the inner function with respect to x. ie #(du)/dx#, where u #=sqrt(x-3)# That is, #d/(dx) f(g(x)) = f'(g(x))*g'(x)#

The derivative of #cot(x)# with respect to #x# is #-csc^2(x)#, with respect to #g(x)# this becomes #-csc^2(sqrt(x-3))#, the derivative of #sqrt(x-3)# is #0.5(sqrt(x-3))^-(1/2)# which is # = 1/(2*sqrt(x-3))#

Thus, we obtain the answer by multiplying the two, so #-csc^2(sqrt(x-3)) * 1/(2*sqrt(x-3)) = -csc^2(sqrt(x-3))/(2*sqrt(x-3))#