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): $\frac{sin(x+a)}{cos\,a}$

Answer 1

Let f(x)= $\frac{sin(x+a)}{cos\,a}$
By the quotient rule,
f'(x)=$\frac{cos\,x\frac{d}{dx}[sin(x+a)]-sin(x+a)\frac{d}{dx}cos\,x}{cos^2x}$
f'(x)=$\frac{cos\,x\frac{d}{dx}[sin(x+a)]-sin(x+a)(-sin\,x)}{cos^2x}$ ...(i)
Let g(x)=sin(x+a). Accordingly, g(x+h)=sin(x+h+a)
By first principle,
g'(x)= $\lim_{h\rightarrow 0}$ $\frac{g(x+h)-g(x)}{h}$
= $\lim_{h\rightarrow 0}$ $\frac{1}{h}$ [sin(x+h+a) - sin(x+a)]
= $\lim_{h\rightarrow 0}$ $\frac{1}{h}$[ 2cos($\frac{(2x+2h+h)}{2}$) sin($\frac{h}{2}$)]
= $\lim_{h\rightarrow 0}$ [cos($\frac{(2x+2h+h)}{2}$)$\frac{sin(\frac{h}{2})}{(\frac{h}{2})}$]
=cos$\frac{(2x+2a)}{2}$ $\times$1
= cos(x+a) ...(ii)
From (i) and (ii), we obtain
f'(x)= $\frac{cos\,x.cos(x+a)+sin\,xsin(x+a)}{cos^2x}$
=$\frac{cos(x+a-x)}{cos^2x}$
=$\frac{cos\,a}{cos^2\,x}$