# Find the derivative of the following functions

Question:

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

Solution:

Let $f(x)=\sin (x+a)$

$f(x+h)=\sin (x+h+a)$

By first principle,

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$=\lim _{h \rightarrow 0} \frac{\sin (x+h+a)-\sin (x+a)}{h}$

$=\lim _{h \rightarrow 0} \frac{1}{h}\left[2 \cos \left(\frac{x+h+a+x+a}{2}\right) \sin \left(\frac{x+h+a-x-a}{2}\right)\right]$

$=\lim _{h \rightarrow 0} \frac{1}{h}\left[2 \cos \left(\frac{2 x+2 a+h}{2}\right) \sin \left(\frac{h}{2}\right)\right]$

$=\lim _{h \rightarrow \infty}\left[\cos \left(\frac{2 x+2 a+h}{2}\right)\left\{\frac{\sin \left(\frac{h}{2}\right)}{\left(\frac{h}{2}\right)}\right\}\right]$

$=\lim _{h \rightarrow 0} \cos \left(\frac{2 x+2 a+h}{2}\right) \lim _{\frac{a}{2} \rightarrow 0}\left\{\frac{\sin \left(\frac{h}{2}\right)}{\left(\frac{h}{2}\right)}\right\}$ $\left[\right.$ As $\left.h \rightarrow 0 \Rightarrow \frac{h}{2} \rightarrow 0\right]$

$=\cos \left(\frac{2 x+2 a}{2}\right) \times 1$ $\left[\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right]$

$=\cos (x+a)$