Derivative of tan^-1(x^3)

Step by step

The question

Question: find the derivative of f(x) = tan⁻¹(x³).

1 The rule

The rule: the derivative of tan⁻¹ u is 1/(1 + u²).

2 The inside

There is something inside: x³.

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 x³ gives 3x².

5 Multiply

The chain rule: multiply the two, and put x³ back in place of u.

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 3x².

Square the inside: (x³)² is x⁶.

Tidy up: f′(x) = 3x² over (1 + x⁶).

6 Check

Let's check: the derivative is the slope.

At x = 1, our answer gives 1.5, with a calculator in radians.

Now the slope between two very close points: x = 1 and x = 1.001.

A calculator gives about 1.5.

Very close to 1.5.

Right!

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.