How do you find the 50th derivative of y=cos(x) ?

1 Answer
Aug 1, 2014

First, it's recommended to obtain a formula for the nth derivative of cosx.

To do this, usually it is needed to continually differentiate until you notice a pattern.

So we will begin by taking the first derivative:

dy/dx = -sinx

Next, the second derivative:

(d^2y)/(dx^2) = -cosx

And the third derivative:

(d^3y)/(dx^3) = sinx

The fourth:

(d^4y)/(dx^4) = cosx

There - we've arrived back at cosx. Since this was our original function, differentiating again will just give us the first derivative, and so on. So, we can deduce that the nth derivative is periodic.

Now the problem is putting this pattern into a formula. At first it might look like there's no mathematically explainable pattern - we have a negative, then a negative, then a positive, then a positive, meanwhile flipping from sine to cosine - but when you graph these successive functions, it's easy to see that each graph is the previous derivative, but shifted to the left by pi/2.

What do I mean? Well, -sin x is the same thing as cos(x + pi/2). And -cos x is the same thing as cos(x + pi).

So there's our formula:

f^n(x) = cos(x + (pi n)/2)

Now, if we substitute n = 50, we obtain:

f^50(x) = cos(x + 25pi)

Since cosine itself is periodic, we can divide the 25 by 2 and leave the remainder next to the pi:

f^50(x) = cos(x + pi)

Which is the same thing as:

f^50(x) = -cosx