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

##### 1 Answer

First, it's recommended to obtain a formula for the

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

So we will begin by taking the first derivative:

Next, the second derivative:

And the third derivative:

The fourth:

There - we've arrived back at

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

What do I mean? Well,

So there's our formula:

Now, if we substitute

Since cosine itself is periodic, we can divide the

Which is the same thing as: