Differentiation Of Inverse Function

[Air]

Homework Statement

\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?

 

[chingcx]

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