What is the derivative of #y=arctan(4x)#?
First recall that
Via the chain rule:
If it isn't clear why
We will begin simply with
From this it is implied that
Using implicit differentiation, taking care to use the chain rule on
Which further simplifies to:
Next, a substitution using our initial equation will give us:
This might not look too helpful, but there is a trigonometric identity that can help us.
This simplifies to:
Now, simply multiply a few things around, and we get:
Beautiful. Now we can simply substitute into the equation we have in step 6:
And voilà - there's our identity.