Find the derivative of the function given by f(x) = (x^2). - Vidyadip

Question

Find the derivative of the function given by $f(x) = \sin (x^2).$

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Answer

Observe that the given function is a composite of two functions. Indeed, if $t = u(x) = x^2$ and $v(t) = \sin t,$ then
$f(x) = (v\ o\ u) (x) = v(u(x)) = v(x^2) = \sin x^2$
Put $t = u(x) = x^2.$ Observe that $\frac{d v}{d t}$ = cost and $\frac{d t}{d x} = 2x$ exist.
Hence, by chain rule
$\frac{d f}{d x}=\frac{d v}{d t} \cdot \frac{d t}{d x} = \cos t.2x$
It is normal practice to express the final result only in terms of $x$. Thus
$\frac{d f}{d x} = \cos t. 2x = 2x \cos x^2$

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