# How do you find the derivative #f(x)=arctan(x/alpha)#?

##### 1 Answer

Jan 6, 2017

#### Explanation:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you can use the chain rule.

#y=arctan(x/alpha) <=> tany=x/alpha #

Differentiate Implicitly:

# sec^2ydy/dx = 1/alpha # ..... [1]

Using the

# tan^2y+1 = sec^2y #

# :. (x/alpha)^2+1=sec^2y #

Substituting into [1]

# ((x/alpha)^2+1)dy/dx=1/alpha #

# :. ((x^2+alpha^2)/alpha^2)dy/dx=1/alpha #

# :. dy/dx=1/alpha * alpha^2/(x^2+alpha^2)#

# :. dy/dx=alpha/(x^2+alpha^2)#