# 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 ) $\sin (x+a)$

Let $f(x)=\sin (x+a)$
$f(x+h)= \sin (x+h+a)$
By first principle,
$f'(x) = \lim\limits_{h \to 0} \large\frac{f(x+h)-f(x)}{h}$
$= \lim\limits_{h \to 0} \large\frac{ \sin (x+h+a)-\sin (x+a)}{h}$
$= \lim\limits_{ h \to 0}\large\frac{1}{h}$$\bigg[ 2 \cos \bigg( \Large\frac{x+h+a+x+a}{2} \bigg)$$\sin\bigg( \Large\frac{x+h+a-x-a}{2} \bigg) \bigg]$
$= \lim\limits_{ h \to 0}\large\frac{1}{h}$$\bigg[ 2 \cos \bigg(\large\frac{2x+2a+h}{2}\bigg)$$ \sin\bigg( \Large\frac{h}{2}\bigg) \bigg]$
$\lim\limits_{h \to 0} \bigg[\cos \bigg( \Large\frac{2x+2a+h}{2} \bigg) \bigg \{ \Large\frac{\sin \bigg( \Large\frac{h}{2} \bigg)}{ \bigg( \Large\frac{h}{2} \bigg)} \bigg\} \bigg]$
$\lim\limits_{h \to 0} \cos \bigg( \Large\frac{2x+2a+h}{2} \bigg)$$\lim\limits_{ \large\frac{h}{2} \to 0} \bigg \{ \Large\frac{\sin \bigg( \Large\frac{h}{2} \bigg)}{ \bigg( \Large\frac{h}{2} \bigg)} \bigg\}$$\qquad \bigg[ As \: h \to 0 \Rightarrow \large\frac{h}{2} $$\to 0\bigg] = \cos \bigg( \large\frac{2x+2a}{2} \bigg)$$ \times 1$ $\qquad \bigg[ \lim\limits_{ x \to 0} \large\frac{\sin x}{x}$$=1 \bigg]$
$= \cos ( x+a)$