Find the derivative of the following function | vedclass

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Question

(A) Let $f(x) = \sin(x+1)$. Accordingly,$f(x+h) = \sin(x+h+1)$.By the first principle of derivatives:$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Vedclass · Accepted Answer

$\cos(x+1)$

Vedclass · Answer

(A) Let $f(x) = \sin(x+1)$. Accordingly,$f(x+h) = \sin(x+h+1)$.By the first principle of derivatives:$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$= \lim_{h 	o 0} \frac{\sin(x+h+1) - \sin(x+1)}{h}$Using the identity $\sin A - \sin B = 2 \cos(\frac{A+B}{2}) \sin(\frac{A-B}{2})$:$= \lim_{h 	o 0} \frac{2 \cos(\frac{x+h+1+x+1}{2}) \sin(\frac{x+h+1-x-1}{2})}{h}$$= \lim_{h 	o 0} \frac{2 \cos(\frac{2x+h+2}{2}) \sin(\frac{h}{2})}{h}$$= \lim_{h 	o 0} [\cos(\frac{2x+h+2}{2}) \cdot \frac{\sin(h/2)}{h/2}]$$= \cos(\frac{2x+2}{2}) \cdot 1$$= \cos(x+1)$

Find the derivative of the following function from first principles: $\sin(x+1)$

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