Answer to Question #204421 in Calculus for Kenneth

Question

Using first principle show that the derivative of COSX = -SINX

Answer 1

$We \space will \space find \space derivative \\ \space of \space cos (x) with \space first \space principle.\\ First \space principle \space formula \space of \\ \space derivative \space such \space that...\\ f'(x)=\\\displaystyle \lim_{h\rightarrow 0}\;\;\;\frac{f(x+h)-(x)}{h}\\\\ =\displaystyle \lim_{h\rightarrow 0}\;\;\;\frac{cos(x+h)-cos(x)}{h}\\\\ =\displaystyle \lim_{h\rightarrow 0}\;\;\;\frac{cosx*cosh-sinx*sinh-cos(x)}{h}\\\\ =\displaystyle \lim_{h\rightarrow 0}\;\;\;\frac{cosx(cosh-1)-sinx*sinh}{h}\\\\ =\displaystyle \lim_{h\rightarrow 0}\;\;\;\frac{cosx(cosh-1)}{h}-\frac{sinx*sinh}{h}\\\\ =\displaystyle \lim_{h\rightarrow 0}\;\;\;cosx*\frac{(cosh-1)}{h}-sinx*\frac{sinh}{h}\\\\ =cosx*0-sinx*1\\\\ =0-sin(x)\\\\ =-sin(x)$