Answer
$cosx$
Work Step by Step
$\frac{d}{dx}[cosx]=-sinx$
$\frac{d^{2}}{d^{2}x}[cosx]=-cosx$
$\frac{d^{3}}{d^{3}x}[cosx]=sinx$
$\frac{d^{4}}{d^{4}x}[cosx]=cosx$
Here we return to the original function, so the pattern will repeat each 4 derivatives. For example, the multiples of 4 will be just like the 4th derivative: 8, 12, 16,... And the ones before them will be just like the 3rd derivative: 7, 11, 15...
We want the 100th derivative, 100 is a multiple of 4, since 4•25=100, which means:
$\frac{d^{100}}{d^{100}x}[cosx]=\frac{d^{4}}{d^{4}x}[cosx]=cosx$
