Step by step
The question
Question: find the derivative of f(x) = 2tan⁻¹ x.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The rule
The rule: the derivative of tan⁻¹ x is 1/(1 + x²).
Careful!
The little −1 means the inverse function, the angle.
It does not mean 1 over tan x.
2 Use the rule
The 2 in front just waits, and multiplies.
So f′(x) = 2/(1 + x²).
3 Check
Let's check at one x.
Any x works: take x = 1.
The derivative is the slope there.
Our answer gives 1, with a calculator in radians.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 1.
Very close to 1.
Right!
4 Pick the choice
Our answer is f′(x) = 2/(1 + x²).
That is choice A.
5 Why not the others
B is 2/√(1 − x²).
Wrong: that is the rule for sin⁻¹ x.
C is 1/(1 + x²).
Wrong: lost the 2.
D is −2/(1 + x²).
Wrong: the minus belongs to cos⁻¹ x.
Let's look back.
First: the rule.
Then: use the rule.
Then: check.
Then: pick the choice.
Last: why not the others.