Question

What is the derivative and second derivative of `cos(-x)`?

Answer 1

`d/dxcos(-x)=\color(blue)(sin(-x))` and `d/(d^2x)cos(-x)=\color(green)(-cos(-x))`

Explanation:

`d/dxcos(-x)=d/dx(-x)d/(d(-x))cos(-x)`
..... used the chain rule
`d/dxcos(-x)=-1*-sin(-x)=\color(blue)(sin(-x)`
..... derivative of `cos(x)=-sin(x)`

`d/(d^2x)cos(-x)=d/dx(d/dxcos(-x))=d/dx(\color(blue)(sin(-x)))`
..... second derivative is derivative of first derivative
`d/(d^2x)cos(-x) = d/dx(-x)d/(d(-x))sin(-x)`
..... used the chain rule again
`d/(d^2x)cos(-x)=-1*cos(-x) = \color(green)(-cos(-x)`
..... derivative of `sin(x)=cos(x)`