Derivate Of y=cosx Proof

Prove that:

If\quad f(x)=cosx,\quad f

\lim _{ \delta x\rightarrow 0 }{ \frac { f\left( x+\delta x \right) -f\left( x \right) }{ \delta x } } \\ \\ =\lim _{ \delta x\rightarrow 0 }{ \frac { cos\left( x+\delta x \right) -cosx }{ \delta x } } \\ \\ =\lim _{ \delta x\rightarrow 0 }{ \frac { cosxcos\delta x-sinxsin\delta x-cosx }{ \delta x } } \\ \\ =\lim _{ \delta x\rightarrow 0 }{ \frac { cosx\left( cos\delta x-1 \right) -sinxsin\delta x }{ \delta x } } \\ \\ =\lim _{ \delta x\rightarrow 0 }{ \frac { cosx\left( cos\delta x-1 \right) }{ \delta x } } -\frac { sinxsin\delta x }{ \delta x } \\ \\ =0-sinx=-sinx\\ \\ As:\\ \\ \lim _{ \delta x\rightarrow 0 }{ \frac { \left( cos\delta x-1 \right) }{ \delta x } } =0\\ \\ \lim _{ \delta x\rightarrow 0 }{ \frac { sin\delta x }{ \delta x } } =1

Other posts you may be interested in...