Find the derivative of the function given by f(x) = sin (x2). - Vidyadip

Question

Find the derivative of the function given by f(x) = sin (x²).

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Answer

Observe that the given function is a composite of two functions. Indeed, if t = u(x) = x² and v(t) = sin t, then
f(x) = (v o u) (x) = v(u(x)) = v(x²) = sin x²
Put t = u(x) = x². Observe that $\frac{d v}{d t}$ = cost and $\frac{d t}{d x}$ = 2x exist. Hence, by chain rule
$\frac{d f}{d x}=\frac{d v}{d t} \cdot \frac{d t}{d x}$ = cost.2x
It is normal practice to express the final result only in terms of x. Thus
$\frac{d f}{d x}$ = cost. 2x = 2x cos x²

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