Find the derivative of the function given by | vedclass

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Question

(A) The given function is a composite function $f(x) = v(u(x))$,where $u(x) = x^{2}$ and $v(t) = \sin(t)$.By the chain rule,the derivative is given by

Vedclass · Accepted Answer

$2x \cos(x^{2})$

Vedclass · Answer

(A) The given function is a composite function $f(x) = v(u(x))$,where $u(x) = x^{2}$ and $v(t) = \sin(t)$.By the chain rule,the derivative is given by $\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}$.Here,$\frac{dv}{dt} = \frac{d}{dt}(\sin t) = \cos t$ and $\frac{dt}{dx} = \frac{d}{dx}(x^{2}) = 2x$.Substituting these values,we get $\frac{df}{dx} = \cos(t) \cdot 2x$.Replacing $t$ with $x^{2}$,we obtain $\frac{df}{dx} = 2x \cos(x^{2})$.

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

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