y = tan^-1(4x^2), find dy/dx

Step by step

The question

Question: find dy/dx for y = tan⁻¹(4x²).

1 The rule

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

2 The inside

There is something inside: 4x².

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 4x² gives 8x.

5 Multiply

The chain rule: multiply the two, and put 4x² back in place of u.

Careful!

Do not stop at the outside.

Multiply by the inside's slope, 8x.

Square the inside: (4x²)² is 16x⁴.

Tidy up: dy/dx = 8x/(1 + 16x⁴).

6 Check

Let's check: the derivative is the slope.

At x = 1, our answer gives 0.4706, with a calculator in radians.

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

A calculator gives about 0.47.

Very close to 0.4706.

Right!

Let's look back.

First: the rule.

Then: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.