What is the derivative of this function #y=4tan^-1(3x^4)#?

The first step to any intermediate derivative problem (like this one) is to identify what rules you'll need to use to solve it. In this, you have a composition of functions (that is, one function embedded inside another), which means you'll need to use a chain rule at some point.

Now, onto the actual computations. It'd be smart to pull that 4 out of your calculations, leaving the following:

#4 * d/dx[tan^-1(3x^4)]#

What this does is it allows you to focus on the essential functions, which will greatly reduce your chances of making a mistake.

Now, you take the derivative of #[tan^-1(3x^4)]#. Because you have that #3x^4# embedded into the inverse tangen function, you'll need to invoke the chain rule (as alluded to earlier) to solve this problem. This is as follows:

#d/dx(f(g(x)) = f'(g(x)) * g'(x)#

In English, you take the derivative of your outermost function without touching the inner function, and then multiply that by the derivative of the inner function.

Our outer function is clearly #tan^-1(x)#, and the inner function is #3x^4#. Following the form of the chain rule gives:

#1/(1 + (3x^4)^2) * 12x^3#

Let's not forget that 4 we pulled out earlier! Putting that back in, as well as cleaning up the expression a bit, leaves:

#(48x^3)/(1 + 9x^8)#

That's how you'd solve this. This was a pretty straightforward chain rule problem, but in the future you'll need to be able to work with much more dense functions, and be able to break them apart to calculate derivatives.

If you'd like some additional resources, I have made a theory video , as well as a practice problem video on the Chain Rule, and I'd encourage you to check those out.

Hope that helped :)