Home > Calculus > Chain Rule

How do you differentiate #f(x)=1/sin(sqrt(1/x))# using the chain rule?

Answer 1

Ezra Adams

#f'(x)=(cos(1/sqrtx))/(2sin^2(1/sqrtx)x^(3/2))#

#f(x)=(sin(x^(-1/2)))^-1#

Use the chain rule: #d/dxu^-1=-u^-2*(du)/dx#

#f'(x)=-sin^-2(x^(-1/2))*color(blue)(d/dxsin(x^(-1/2))#

To find this derivative, use chain rule again:

#d/dxsin(u)=cos(u)*(du)/dx#

Thus,

#color(blue)(d/dxsin(x^(-1/2)))=cos(x^(-1/2))*d/dx x^(-1/2)#

#=>color(blue)(-1/2x^(-3/2)cos(x^(-1/2))#

Plug this back in.

#f'(x)=-sin^-2(x^(-1/2))*-1/2x^(-3/2)cos(x^(-1/2))#

#=>(cos(1/sqrtx))/(2sin^2(1/sqrtx)x^(3/2))#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Trending questions

Not the question you need?

HIX Tutor

Solve ANY homework problem with a smart AI

98% accuracy study help
Covers math, physics, chemistry, biology, and more
Step-by-step, in-depth guides
Readily available 24/7