Question

Find the derivative of cos(2x-3) from first principle

Answer 1

-2sin(2x-3)

Hope this helps

Answer 2

The derivative of cos(2x-3) from first principle is -2sin(2x-3)

• From first principle, the derivative of a function is given by

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

           Here,

                 f(x) = cos(2x-3)

                 f(x+h) = cos(2x + 2h -3)

• It's derivative from first principle is

                 $f'(x) =  \lim_{h \to \ 0} \frac{cos(2x+2h-3)-cos(2x-3)}{h} \\f'(x) =  \lim_{h \to \ 0} \frac{-2sin(\frac{2x+2h-3+2x-3}{2})sin(\frac{2x+2h-3-2x+3}{2} ) }{h} \\cosC-cosD=-2sin(\frac{C+D}{2})sin( \frac{C-D}{2})$

                 $f'(x)=-2 \lim_{h \to \ 0} \frac{sin(2x+h-3)sinh}{h} \\f'(x)=-2 \lim_{h \to \ 0}sin(2x+h-3) \lim_{h \to \ 0}\frac{sinh}{h} \\f'(x)=-2sin(2x-3)$