Question Video: The Derivative of an Inverse Tangent Function Mathematics • Higher Education

Find (d/d𝑥) tan⁻¹ 𝑥.

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Video Transcript

Find the derivative of the inverse of tan 𝑥.

So, the first thing I’m gonna do is let 𝑓 of 𝑥 equal the inverse of tan 𝑥. So then, if I take the tan of both sides of the equation, I’m gonna get tan of 𝑓 of 𝑥 is equal to 𝑥. And that’s because if you take tan of inverse tan of 𝑥, then this is just equal to 𝑥. It’s also worth noting at this point something that’s gonna be useful later on. And that is that the first derivative of 𝑓 of 𝑥 must be equal to the first derivative of inverse tan 𝑥 because we already said that 𝑓 of 𝑥 is equal to inverse tan 𝑥.

So now, if we say let’s look at the derivative of both sides of our equation, we can say that the derivative of tan 𝑓 of 𝑥 is equal to the derivative of 𝑥. So now, what we can do is use one of our standard derivatives to help us work out what the derivative of tan 𝑓 of 𝑥 is going to be. And that’s because if we take the derivative of tan 𝑎𝑥, this is equal to 𝑎 sec squared 𝑎𝑥. And this is where the 𝑎 before the sec squared 𝑎𝑥 is just the result of finding the derivative of 𝑎𝑥. So therefore, it’s gonna be equal to sec squared 𝑓 of 𝑥 multiplied by the derivative of 𝑓 of 𝑥, and this is equal to one. And that’s because the derivative of 𝑥 is just one.

And now, what we can do is substitute in the inverse of tan 𝑥 for 𝑓 of 𝑥. Because we know from the beginning that’s what they’re equal to. So, therefore, what we’re gonna get is sec squared inverse of tan 𝑥 multiplied by the derivative of 𝑓 of 𝑥 is equal to one. So, therefore, if we rearrange this, we can say that the derivative of 𝑓 of 𝑥 is equal to one over sec squared inverse tan of 𝑥.

So now, what we can do is use one of our trig identities. And that one is that sec squared 𝑥 is equal to tan squared 𝑥 plus one. And if we substitute that in, so when we substitute this in, we get the derivative of 𝑓 of 𝑥 is equal to one over tan squared the inverse of tan 𝑥 plus one. So, therefore, we’re gonna have the derivative of 𝑓 of 𝑥 is gonna be equal to one over 𝑥 squared plus one. That’s cause tan squared of the inverse tan of 𝑥 gives us 𝑥 squared.

So, therefore, the derivative of the inverse of tan 𝑥 is gonna be equal to one over 𝑥 squared plus one.