Question

Find the derivative of $\sin (x+a)$ where $a$ is a fixed non-zero constant.

Answer 1

Let $f\left(x\right)=\sin (x+a)$
Using $\sin (A+B)=\sin A\cos B+\cos A\sin B$
$f\left(x\right)=\sin x\cdot \cos a+\cos x\cdot \sin a$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}(\sin x\cdot \cos a+\cos x\cdot \sin a)$
$\Rightarrow f^{\prime }\left(x\right)=\cos a\frac{d}{dx}\left(\sin x\right)+\sin a\frac{d}{dx}\left(\cos x\right)$
$\Rightarrow f^{\prime }\left(x\right)=\cos a\cos x+\sin a(-\sin x)$
$\Rightarrow f^{\prime }\left(x\right)=\cos a\cos x-\sin a\sin x$
Using $\cos (A+B)=\cos A\cos B-\sin A\sin B$
$\therefore f^{\prime }\left(x\right)=\cos (x+a)$