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): sin (x + a)

Asked by Pragya Singh | 1 year ago |  71

##### Solution :-

Let, f(x) = sin(x + a) f(x + h) = sin(x + h + a)

By first principle,

$$f'(x)=\lim\limits_{h \to 0}\dfrac{f(x+h)-f(x)}{h}$$

$$\lim\limits_{h \to 0}\dfrac{sin(x+h+a)-sin(x+a)}{h}$$

$$\lim\limits_{h \to 0}\dfrac{1}{h}[2cos(\dfrac{x+h+a+x+a}{2})$$

$$sin(\dfrac{x+h+a-x-a}{2})]$$

$$\lim\limits_{h \to 0}\dfrac{1}{h}[2cos(\dfrac{2x+2a+h}{2})sin(\dfrac{h}{2})]$$

$$\lim\limits_{h \to 0}[cos(\dfrac{2x+2a+h}{2})[\dfrac{sin(\dfrac{h}{2})}{(\dfrac{h}{2})}]]$$

$$\lim\limits_{h \to 0}cos(\dfrac{2x+2a+h}{2}). \lim\limits_{\dfrac{h}{2} \to 0}[\dfrac{sin(\dfrac{h}{2})}{(\dfrac{h}{2})}]$$

$$cos( \dfrac{2x+2a}{2})\times 1$$

=cos (x + a)

Answered by Abhisek | 1 year ago

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