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 non-zero constants and $m$ and $n$ are integers) : $\frac{\sin (x+a)}{\cos x}$

Answer 1

Let $f\left(x\right)=\frac{\sin (x+a)}{\cos x}$
Thus by quotient rule,
$f^{\prime }\left(x\right)=\frac{\cos x\frac{d}{dx}\left[\sin (x+a)\right]-\sin (x+a)\frac{d}{dx}\cos x}{{\cos }^{2}x}$
$=\frac{\cos x\cos (x+a)-\sin (x+a)(-\sin x)}{{\cos }^{2}x}$
$=\frac{\cos x.\cos (x+a)+\sin x\sin (x+a)}{{\cos }^{2}x}$
$=\frac{\cos (x+a-x)}{{\cos }^{2}x}=\frac{\cos a}{{\cos }^{2}x}$