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

The chain rule is used whenever there are functions of functions. In other words, when you have a function of the form;

#f(x)=g(h(x))#

The chain rule tells us that the derivative of this function is;

#f'(x)=g'(h(x))*h'(x)#

The first step to using the chain rule is to identify the separate functions. Lets start by rewriting the given function to make the component functions easier to identify.

#1/cos(sqrt(1/x)) = cos^-1((1/x)^(1/2))=cos^-1(x^(-1/2))#

Now we have a function of the form;

#f(g(h(x)))#

Where;

#f(x)=x^-1#
#g(x)=cosx#
#h(x)=x^(-1/2)#

You can see that if you work backwards plugging these functions into each other, you get the original function back.

#f(g(h(x)))=f(g(x^(-1/2)))) = f(cos(x^(-1/2))) = cos^-1(x^(-1/2))#

We can take the derivative of each of these component functions to get;

#f'(x)=-x^-2#
#g'(x)=-sinx#
#h'(x)=-x^(-3/2)/2#

Now we apply the chain rule to our function.

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

Apply the chain rule again.

#=-cos^-2(x^(-1/2))(-sin(x^(-1/2)))d/dx x^(-1/2)#

Take the last derivative.

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

Simplify.

#=-(sec(1/sqrt(x)) tan(1/sqrt(x)))/(2sqrt(x^3)) #