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

Question

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

Evelyn Agard · Accepted Answer

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

Charles Arocha · Answer

undefined To differentiate $ f(x) = \cot(\sqrt{x - 3}) $ using the chain rule:

1. Identify the outer function and the inner function.
   - Outer function: $ \cot(u) $
   - Inner function: $ u = \sqrt{x - 3} $

2. Differentiate the outer function with respect to its variable.
   - $ \frac{d}{du} \cot(u) = -\csc^2(u) $

3. Differentiate the inner function with respect to the variable $ x $.
   - $ \frac{du}{dx} = \frac{1}{2\sqrt{x - 3}} $

4. Apply the chain rule, multiplying the derivatives obtained in steps 2 and 3.
   - $ \frac{df}{dx} = \frac{d}{dx} \cot(\sqrt{x - 3}) = -\csc^2(\sqrt{x - 3}) \cdot \frac{1}{2\sqrt{x - 3}} $

5. Simplify the expression if needed, resulting in the final derivative.