Step by step
The question
Question: find the derivative of f(x) = tan⁻¹(3x).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The rule
The rule, with an inside u: tan⁻¹ u gives 1/(1 + u²).
The chain rule then multiplies by the inside's slope.
2 The inside
There is something inside: 3x.
Call it u.
3 The outside
First the outside: the rule on tan⁻¹ u.
4 The inside's slope
The inside's slope: the power rule on 3x gives 3.
5 Multiply
The chain rule: multiply the two, and put 3x back in place of u.
Square the inside: (3x)² is 9x².
Tidy up: f′(x) = 3/(1 + 9x²).
Careful!
Do not stop at the outside.
Multiply by the inside's slope, 3.
6 Check
Let's check at one x.
Any x works: take x = 1.
The derivative is the slope there.
Our answer gives 0.3, with a calculator in radians.
Now the slope between two very close points: x = 1 and x = 1.001.
A calculator gives about 0.3.
Very close to 0.3.
Right!
7 Pick the choice
Our answer is f′(x) = 3/(1 + 9x²).
That is choice C.
8 Why not the others
A is 1/(1 + 9x²).
Wrong: lost × 3, the derivative of the inside.
That is the Careful mistake from before.
B is 3/(1 + 3x²).
Wrong: did not square the 3: (3x)² is 9x².
D is 3/(1 + x²).
Wrong: left the inside out of the bottom.
Let's look back.
First: the rule.
Then: the inside.
Then: the outside.
Then: the inside's slope.
Then: multiply.
Then: check.
Then: pick the choice.
Last: why not the others.