Question

Find the derivative of the following functions from first principle:
$\sin (x+1)$

Answer 1

Let $f\left(x\right)=\sin (x+1)$
Thus using first principle,
$f^{\prime }\left(x\right)=\underset{x\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
=$\underset{x\to 0}{lim}\frac{1}{h}[\sin (x+h+1)-sin(x+1\left)\right]$
=$\underset{x\to 0}{lim}\frac{1}{h}\left[2\cos \left(\frac{x+h+1+x+1}{2}\right)\sin \left(\frac{x+h+1-x-1}{2}\right)\right]$
=$\underset{x\to 0}{lim}\frac{1}{h}\left[2\cos \left(\frac{2x+h+2}{2}\right)\sin \left(\frac{h}{2}\right)\right]$
=$\underset{x\to 0}{lim}\left[\cos \left(\frac{2x+h+2}{2}\right)\frac{\sin \left(\frac{h}{2}\right)}{\left(\frac{h}{2}\right)}\right]$
=$\cos \left(\frac{2x+0+2}{2}\right)\cdot 1=\cos (x+1)$