988 views

### 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.

3 years ago

Answered by Jonathan, an A Level Further Mathematics tutor with MyTutor

## Still stuck? Get one-to-one help from a personally interviewed subject specialist

#### 122 SUBJECT SPECIALISTS

£20 /hr

Degree: Aeronautical Engineering (Masters) - Imperial College London University

Subjects offered:Further Mathematics , Science+ 3 more

Further Mathematics
Science
Physics
Maths
Biology

“Expert tuition from an expert. A second year Uni student who's completed the hassle of GCSEs and A Levels, and can teach you the easy way to pass and succeed!”

MyTutor guarantee

|  1 completed tutorial

£30 /hr

Degree: MMathPhil Mathematics and Philosophy (Bachelors) - Oxford, St Anne's College University

Subjects offered:Further Mathematics , Maths+ 3 more

Further Mathematics
Maths
.MAT.
-Personal Statements-
-Oxbridge Preparation-

“University of Oxford Maths and Philosophy student happy to help students learn and stay motivated!”

£26 /hr

Degree: Physics and Computer Science (Natural Sciences) (Masters) - Durham University

Subjects offered:Further Mathematics , Science+ 6 more

Further Mathematics
Science
Physics
Maths
Computing
Chemistry
Biology
-Personal Statements-

“About Me: I'm a Physics and Computer Science student at Durham University, which sounds boring but it's actually really fun.I've got a great knack for making boring things more exciting, which is a good job given my degree title!  I'...”

Currently unavailable: for new students

Degree: Physics with Astrophysics (Masters) - Manchester University

Subjects offered:Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“I have been a private tutor for 4 years, initially teaching GCSE Science and Maths and continuing on to A Level Maths and Physics for the last 2 years. I am also a qualified tutor with The Tutor Trust.”

### You may also like...

#### Other A Level Further Mathematics questions

For f(x) = (3x+4)^(-2), find f'(x) and f''(x) and hence write down the Maclaurin series up to and including the term in x^2.

How do I solve a simultaneous equation with more unknowns than equations?

Find the displacement function if the acceleration function is a=2t+5. Assume a zero initial condition of displacement and v=8 when t=1.

Find the inverse of the general 2x2 matrix A= ([a, b],[c, d]) when does this inverse exist?

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this.