I am able to find derivatives of $\sin x$ and $\sin 2x$ using first principle (Using the formula for $\sin(A)-\sin(B)$ and subsequently using $\lim_{x\rightarrow 0}$ $\frac{\sin x}{x}$ = 1. But I am getting stuck in trying to find Derivative of $\sin(x^2)$ using the same.
After using the Sin A - Sin B formula I get the following result but then I am unable to separate out $x$ and $t$ to get a $\frac{\sin(t)}{t}$ form: $$\frac{2\cos(x^2+x\,t+\frac{t^2}{2})\sin(x\,t+\frac{t^2}{2})}{t}$$ and solve it further.
Request Guide.