How do you differentiate arctan(x)?

Differentiation of arctan(x), also written tan^(-1)(x), requires a little knowledge of regular trigonometric algebra and differentiation, and implicit differentiation. 

Let's set y = arctan(x) to start. Then, by the definition the arctan function, tan(y) = x. We recall the derivative of the tan function, d(tan(u))/du = (sec(u))^2.

So, if we differentiate both sides of our equation with respect to x, we find that (dy/dx)((sec(y))^2) = 1, using the rules of implicit differentiation. Now, we can rearrange our equation for dy/dx = 1/((sec(y))^2). 

Thinking about some trigonometric algebra, we know that* (sec(y))^2 = 1 + (tan(y))^2. Additionally, remember that tan(y) = x as we said earlier! Using this information, simple substitution back into our equation yields dy/dx = 1/(1+x^2).

So, d(arctan(x))/dx = 1/(1+x^2).

 

 

 

*For a reminder of where this comes from, think about the trigonometric relation: (sin(y))^2 + (cos(y))^2 = 1.

Related Further Mathematics A Level answers

All answers ▸

A=[5k,3k-1;-3,k+1] where k is a real constant. Given that A is singular, find all the possible values of k.


Give the general solution to (d2y/dx2) - 2dy/dx -3y = 2sinx


How do you find the cube root of z = 1 + i?


By Differentiating from first principles, find the gradient of the curve f(x) = x^2 at the point where x = 2


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2024

Terms & Conditions|Privacy Policy