1.(iv)   Find the derivative of the following functions from first principle: \cos ( x - \pi /8 )

Answers (1)

Given.

f(x)=\cos ( x - \pi /8 )

Now, As we know, The derivative of any function at x is 

f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}

f'(x)=\lim_{h\rightarrow 0}\frac{\cos(x+h-\pi/8)-cos(x-\pi/8)}{h}

f'(x)=\lim_{h\rightarrow 0}\frac{-2\sin\left ( \frac{x+h-\pi/8+x-\pi/8 }{2}\right )\sin\left ( \frac{x+h-\pi/8-x+\pi/8}{2} \right )}{h}

f'(x)=\lim_{h\rightarrow 0}\frac{-2\sin\left ( \frac{2x+h-\pi/4 }{2}\right )\sin\left ( \frac{h}{2} \right )}{h}

f'(x)=\lim_{h\rightarrow 0}\frac{-\sin\left ( \frac{2x+h-\pi/4 }{2}\right )\sin\left ( \frac{h}{2} \right )}{\frac{h}{2}}

f'(x)=\sin\left ( \frac{2x+0-\pi/4}{2} \right )\times 1

f'(x)=-\sin\left (x-\pi/8 \right )

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