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

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Answer 1 · 2017-09-01T05:32:59

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

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)#