MAT1193 – 5f Derivative Rules for Trig function We will only study the rules for the sine and cosine functions: f (x) = cos(x) → f ʹ′(x) = −sin(x)
d
cos(x) = −sin(x)
dx
f (x) = sin(x) → f ʹ′(x) = cos(x) d
sin(x) = cos(x)
dx
We won’t study the derivation of these rules, so you won’t need to know the related stuff on pp. 222-­‐223 and pp. 229-­‐230 in the book. €
There are two basic things that you will need to know. First, it’s pretty easy to remember that the derivative of the cosine involves the sine function and the derivative of the sine function involves a cosine. The tricky part is to remember which one gets the negative sine (it’s the cosine). You can try to remember it any way that you want but I always go back to the circle that helps define the sine and cosine. Now consider the value of the input angle x as shown. At that value both sin(x) and cos(x) are positive. Now imagine that the x value increases by a small positive angle Δx. The point will move just a bit further around the circle. That will cause sin(x) to increase and the cos(x) to decrease. Since cos(x) is decreasing and sin(x) is positive, it must be that d
cos(x) = −sin(x)
dx
€
d
sin(x) = cos(x) dx
Since sin(x) is increasing when cos(x) is positive it must be that €
The second thing that you need to know relates to the second derivative f (x) = cos(x) → f ʹ′ʹ′(x) = −cos(x)
d2
d
d
( −sin(x)) = − (sin(x)) = −cos(x)
2 cos(x) =
dx
dx
dx
f (x) = sin(x) → f ʹ′ʹ′(x) = −sin(x)
d2
d
(cos(x)) = −sin(x)
2 sin(x) =
dx
dx
So for both the sine and the cosine, the second derivative is equal to the negative of the function. €