Differentiate (sin(x))/x using definition of a derivative / first principles?
Hello I have a question I have had soem difficulty with for a long time with no answer I could find. How do you differentiate the sine of x divided by x using first principles? I know to use sin(x+h) =sinxcosh + sinhcosx but I am stuck beyond that
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$\begingroup$When we apply the definition of the derivative and replace $x$ with $x+h$
\begin{align}f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}&=\lim_{h\to 0}\frac{\frac{\sin(x+h)}{x+h}-\frac{\sin{x}}{x}}{h}\\& \end{align}
we immediately notice that taking the limit as $h\to0$ would produce the indeterminate form of $\frac{0}{0}$. Hence, we should apply L'Hôpital's rule and differentiate with respect to $h$ to form
$$\lim_{h\to 0}\frac{(x+h)\cos(x+h)-\sin(x+h)}{(x+h)^2}=\lim_{h\to 0}\frac{x\cos(x+h)+h\cos(x+h)-\sin(x+h)}{x^2+2xh+h^2}$$
which upon taking the limit as $h\to0$ will form
$$\frac{x\cos(x)-\sin(x)}{x^2}$$
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