I'm trying to find the derivative of $x^2\sin x$ using only the limit definition of a derivative. I've tried two approaches, one using the difference quotient and another with the regular $x-a$ formula.
I'm stumped on both approaches and not sure where to go. Maybe I'm on the wrong track completely. The difference quotient gets messy quickly and I can't figure out how to factor out $h$ to get it into a definable form. So then I tried: $$\frac{x^2\sin(x) - a^2\sin(a)}{x-a}$$
Is it possible to apply the trig sum-to-product form to the numerator? I'm really just guessing playing around with identities trying to figure this out. Any tips would be appreciated!
$\endgroup$3 Answers
$\begingroup$\begin{align*} &\dfrac{(x+h)^{2}\sin(x+h)-x^{2}\sin x}{h}\\ &=\dfrac{(x+h)^{2}\sin(x+h)-x^{2}\sin(x+h)+x^{2}\sin(x+h)-x^{2}\sin x}{h}\\ &=\dfrac{((x+h)^{2}-x^{2})\sin(x+h)}{h}+\dfrac{x^{2}(\sin(x+h)-\sin x)}{h}\\ &=\dfrac{(2hx+h^{2})\sin(x+h)}{h}+\dfrac{x^{2}(\sin x\cos h+\cos x\sin h-\sin x)}{h}\\ &=(2x+h)\sin(x+h)+x^{2}\cdot\dfrac{\cos h-1}{h}+x^{2}\cos x\cdot\dfrac{\sin h}{h}\\ &\rightarrow 2x\sin x+x^{2}\cos x. \end{align*}
$\endgroup$ $\begingroup$For limits of products, you always need to add a and subtract. In this case, $a^2\sin x$. Then \begin{align} \frac{x^2\sin x - a^2\sin a}{x-a}&= \frac{x^2\sin x - a^2\sin x}{x-a} + \frac{a^2\sin x- a^2\sin a}{x-a}\\ &= \frac{x^2 - a^2}{x-a}\, \sin x + a^2\,\frac{\sin x - \sin a}{x-a}. \end{align}
$\endgroup$ 2 $\begingroup$\begin{equation} \begin{split} \frac{d}{dx}(x^2\sin )\mid_{x=a}&=\lim_{x\rightarrow a} \frac{x^2\sin x-a^2\sin a}{x-a}\\ &=\lim_{x\rightarrow a} \frac{x^2\sin x-a^2\sin x}{x-a}+\frac{a^2\sin x-a^2\sin a}{x-a} \\ &=\lim_{x\rightarrow a} \sin (x)\cdot (x+a)+a^2\frac{\sin ((x-a)+a)-\sin a}{x-a}\\ &=2a\sin a+a^2\cdot \lim_{x\rightarrow a}\frac{\sin (x-a)\cos a+\cos(x-a)\sin a-\sin a}{x-a}\\ &=2a\sin a+a^2\cdot \lim_{x\rightarrow a}\cos a\cdot \frac{\sin(x-a)}{x-a}+\cos a\cdot\frac{\cos(x-a)-1}{x-a}\\ &=2a\sin a+a^2(\cos a\cdot 1+\cos a\cdot0)\\ &=2a\sin a+a^2\cos a\\ \end{split} \end{equation}
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