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

Nov 8, 2015

$- \frac{\sec \left(\frac{1}{\sqrt{x}}\right) \tan \left(\frac{1}{\sqrt{x}}\right)}{2 \sqrt{{x}^{3}}}$

Explanation:

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

$f \left(x\right) = g \left(h \left(x\right)\right)$

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

$f ' \left(x\right) = g ' \left(h \left(x\right)\right) \cdot h ' \left(x\right)$

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.

$\frac{1}{\cos} \left(\sqrt{\frac{1}{x}}\right) = {\cos}^{-} 1 \left({\left(\frac{1}{x}\right)}^{\frac{1}{2}}\right) = {\cos}^{-} 1 \left({x}^{- \frac{1}{2}}\right)$

Now we have a function of the form;

$f \left(g \left(h \left(x\right)\right)\right)$

Where;

$f \left(x\right) = {x}^{-} 1$
$g \left(x\right) = \cos x$
$h \left(x\right) = {x}^{- \frac{1}{2}}$

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;

$f ' \left(x\right) = - {x}^{-} 2$
$g ' \left(x\right) = - \sin x$
$h ' \left(x\right) = - {x}^{- \frac{3}{2}} / 2$

Now we apply the chain rule to our function.

$\frac{d}{\mathrm{dx}} {\cos}^{-} 1 \left({x}^{- \frac{1}{2}}\right) = - {\cos}^{-} 2 \left({x}^{- \frac{1}{2}}\right) \frac{d}{\mathrm{dx}} \cos \left({x}^{- \frac{1}{2}}\right)$

Apply the chain rule again.

$= - {\cos}^{-} 2 \left({x}^{- \frac{1}{2}}\right) \left(- \sin \left({x}^{- \frac{1}{2}}\right)\right) \frac{d}{\mathrm{dx}} {x}^{- \frac{1}{2}}$

Take the last derivative.

$= - {\cos}^{-} 2 \left({x}^{- \frac{1}{2}}\right) \left(- \sin \left({x}^{- \frac{1}{2}}\right)\right) \left(- {x}^{- \frac{3}{2}} / 2\right)$

Simplify.

$= - \frac{\sec \left(\frac{1}{\sqrt{x}}\right) \tan \left(\frac{1}{\sqrt{x}}\right)}{2 \sqrt{{x}^{3}}}$