Whats the derivative forumla for cos(u)
Doing -sin(u) doesnt seem to work.
For eg. Derivative of cos(-2x)
There is a subtlety about differentiation that isn't always stressed when teaching it.
Consider the equation
Here, x appears three times: once in the denominator, and once as the argument of sine and cosine.
Why does this next equation fail?
It's because it doesn't match the pattern above. We've swapped x for 2x in two out of three places. The denominator still says x, however. If you use a chain rule and sub in u=2x, then the derivative you're looking for then looks like the above pattern, with the three x's.
So, when calculating something like
the first step is to make this look something like the derivative rule for sine. However, the argument of the sine function here, which is a cosine, isn't x, so we have to use a chain rule. Let u = cos x. Then, by the chain rule,
It's important to understand what the chain rule does. It essentially changes the derivative from d/dx to d/du, which then allows you to apply the familiar derivative rules. THIS is why we have a chain rule.