How do you differentiate f(x)=sqrt(e^(cot(1/x) using the chain rule.?

1 Answer
Sep 1, 2017

f'(x)=(sqrt(e^(cot(1/x))) csc^2(1/x))/(2 x^2)

Explanation:

Firstly, differentiate the outermost function, which is sqrt(...)

d/dx(sqrt(x))=1/(2sqrt(x))

Hence, d/dx(sqrt(e^(cot(1/x))))=1/(2sqrt(e^(cot(1/x))))*d/dx(e^(cot(1/x)))

Now apply the chain rule again to find d/dx(e^(cot(1/x)))

d/dx(e^x)=e^x

d/dx(e^(cot(1/x)))=e^(cot(1/x))*d/dx(cot(1/x))

Now apply the chain rule one more time

d/dx(cot(x))=-csc^2(x)

d/dx(cot(1/x))=-csc^2(1/x)*d/dx(1/x)

Finally, find the derivative of the innermost function

d/dx(1/x)=-1/x^2

Now just put everything together

d/dx(sqrt(e^(cot(1/x))))=1/(2sqrt(e^(cot(1/x)))) * e^(cot(1/x)) * (-csc^2(1/x)) * (-1/x^2)

After simplifying,

d/dx(sqrt(e^(cot(1/x))))=(sqrt(e^(cot(1/x))) csc^2(1/x))/(2 x^2)