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

1 Answer
kumail
Mar 28, 2017

-csc^2(sqrt(x-3))/(2*sqrt(x-3))−csc2(√x−3)2⋅√x−3

Explanation:

Differentiate the outer function with respect to the inner function (ie dy/(du)dydu, where du = sqrt(x-3)=√x−3 and multiply with the derivative of the inner function with respect to x. ie (du)/dxdudx, where u =sqrt(x-3)=√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))

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