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 nonzero constants and m and n are integers): sin (x + a)

Answer 1

Let f(x)=sin(x+a)
f(x+h)=sin(x+h+a)
By first principle,
f'(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}
= $\lim_{h\rightarrow 0}$ $\frac{1}{h}$[2cos($\frac{x+h+a+x+a}{2}$) sin($\frac{x+h+a-x-a}{2}$)]
= $\lim_{h\rightarrow 0}$ $\frac{1}{h}$[2cos($\frac{2x+2a+h}{2}$) sin($\frac{h}{2}$)]
= $\lim_{h\rightarrow 0}$ [cos($\frac{2x+2a+h}{2}$) $\frac{sin\frac{h}{2}}{\frac{h}{2}}$]
= $\lim_{h\rightarrow 0}$ cos($\frac{2x+2a+h}{2}$) $\lim_{\frac{h}{2}\rightarrow 0}$ {$\frac{sin\frac{h}{2}}{\frac{h}{2}}$} [ As h$\rightarrow$0$\Rightarrow$ \frac{h}{2}$$\rightarrow0]
=cos($\frac{2x+2a}{2}$) x1 [lim x$\rightarrow$0 $\frac{sin\,x}{x}$ =1]
=cos(x+a)