Kpl.2.2

2.2 sinin ja kosinin derivaatat
 
Lause 
D\sin x=\cos x
D\cos x=-\sin x

Esim. Määritä
a) f'(x), kun f(x)=2sinx+cosx
f'\left(x\right)=D2\sin x+D\cos x=2D\sin x+D\sin x=2\cdot\cos x-\sin x
 
b) f'(π/2), kun f(x)=sinx-3cosx
f'\left(x\right)=D\sin x-3D\cos x=\cos x+3\sin x
f'\left(\frac{\pi}{2}\right)=\cos\frac{\pi}{2}+3\sin\frac{\pi}{2}=0+3=3

Esim. Derivoi
a) h(x)=x³cosx
D\left(f\left(x\right)g\left(x\right)\right)=f'\left(x\right)g\left(x\right)+g'\left(x\right)\cdot f\left(x\right)
h'\left(x\right)=3x^2\cos x-x^3\sin x
 
b) h\left(x\right)=\frac{\sin x}{\cos x}
D\frac{f\left(x\right)}{g\left(x\right)}=\frac{f'\left(x\right)g\left(x\right)-g'\left(x\right)f\left(x\right)}{\left(g\left(x\right)\right)^2}\ {,}g\left(x\right)\ne0
h'\left(x\right)=\frac{\cos x\cdot\cos x+\sin x\cdot\sin x}{\left(\cos x\right)^2}=\frac{\cos^2x+\sin^2x}{\cos^2x}=\frac{1}{\cos^2x}