Question

Find the derivative of the following functions (it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $\sin (x+a)$

Answer 1

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