Question

Differentiate the functions with respect to x.
$cos(\sqrt x)$

Answer 1

Let f(x )= $cos(\sqrt x)$
Also, let u(x) = $\sqrt x$
And, v(t) = cos t
Then, vou(x) = v(u(x))
= v($\sqrt x$)
= cos x
= f(x)

Clearly, f is a composite function of two functions, u and v, such that
t = u(x) = $\sqrt x$
Then, $\frac {dt}{dx}$=$\frac {d}{dx}$($\sqrt x$) = $\frac {d}{dx}$($x^\frac 12$)= $\frac12 x^{-\frac 12}$ = $\frac {1}{2\sqrt x}$

And $\frac {dv}{dt}$ = $\frac {d}{dt}$(cos t) = -sin t
=-sin($\sqrt x$)

By using chain rule, we obtain
$\frac {dt}{dx}$ = $\frac {dv}{dt}$ . $\frac {dt}{dx}$
=-sin($\sqrt x$) . $\frac {1}{2\sqrt x}$
=-$\frac {sin\ \sqrt x}{2√x}$

Alternate Method:

\frac {d}{dx}$$[cos (\sqrt x)] = -sin($\sqrt x$) . \frac {d}{dx}$$(\sqrt x)
= -sin($\sqrt x$) . $\frac {d}{dx}$($x^{\frac 12}$)
= -sin$\sqrt x$. \frac 12$$x^{-\frac 12}
= -$\frac {sin\ \sqrt x}{2√x}$