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\(\Rightarrow\) \(\frac{h}{2}\)\(\rightarrow\)0]
=cos(\(\frac{2x+2a}{2}\)) x1 [lim x\(\rightarrow\)0 \(\frac{sin\,x}{x}\) =1]
=cos(x+a)