Differentiation Of Inverse Function

[Air]

Homework Statement

[itex]\frac{\mathrm{d}\left(\frac{1}{a}\tan ^{-1} \left(\frac{x}{a}\right)\right)}{\mathrm{d}x}[/itex]2. The attempt at a solution
Let [itex]y = \frac{1}{a}\tan ^{-1} \left(\frac{x}{a}\right)[/itex]
[itex]\therefore x = a \tan \left(ay\right)[/itex]
Differentiate with respect to [itex]x[/itex] [itex]\rightarrow 1 = a \sec ^2 \left(ay\right) \frac{\mathm{d}y}{\mathrm{d}x}[/itex]
[itex]\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}[/itex]
[itex]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}[/itex]3. The problem I encountered
However, thee answer is incorrect. The correct answer is:
[itex]= \frac{1}{a^2+x^2}[/itex]
Where have I gone wrong?

 

[chingcx]

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