# Differentiation Of Inverse Function

1. Aug 3, 2008

### Air

1. The problem statement, all variables and given/known data
$\frac{\mathrm{d}\left(\frac{1}{a}\tan ^{-1} \left(\frac{x}{a}\right)\right)}{\mathrm{d}x}$

2. The attempt at a solution
Let $y = \frac{1}{a}\tan ^{-1} \left(\frac{x}{a}\right)$
$\therefore x = a \tan \left(ay\right)$
Differentiate with respect to $x$ $\rightarrow 1 = a \sec ^2 \left(ay\right) \frac{\mathm{d}y}{\mathrm{d}x}$
$\therefore \frac{\mathm{d}y}{\mathrm{d}x} = \frac{1}{a \sec ^2 \left(ay\right)} = \frac{1}{a \tan ^2 \left(ay\right) + a}$
$x = a \tan \left(ay\right) \therefore \frac{1}{a \tan ^2 \left(ay\right) + a} = \frac{1}{\left(a \tan \left(ay\right)\right)^2 + a} = \frac{1}{x^2 + a}$

3. The problem I encountered
However, thee answer is incorrect. The correct answer is:
$= \frac{1}{a^2+x^2}$
Where have I gone wrong?

2. Aug 3, 2008

### chingcx

x=a tan(ay)
1= a sec^2(ay) d(ay)/dx