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