What is the second derivative of f(x)=cos(-x)?

Dec 12, 2015

$- \cos \left(- x\right)$

Explanation:

You can compute this in two ways: either you observe that $\cos \left(- x\right) = \cos \left(x\right)$, and so you have

$f \left(x\right) = \cos \left(x\right)$
$f ' \left(x\right) = - \sin \left(x\right)$
$f ' ' \left(x\right) = - \cos \left(x\right)$

Or you use the chain rule and compute

$f \left(x\right) = \cos \left(- x\right)$
$f ' \left(x\right) = - \sin \left(- x\right) \cdot \frac{d}{\mathrm{dx}} \left(- x\right) = - \sin \left(- x\right) \cdot - 1 = \sin \left(- x\right)$
$f ' ' \left(x\right) = \cos \left(- x\right) \cdot \frac{d}{\mathrm{dx}} \left(- x\right) = \cos \left(- x\right) \cdot \left(- 1\right) = - \cos \left(- x\right)$