Derivative of sin(x^2 + 1) Step by Step

Step by step

The question

Question: find dy/dx for y = sin(x² + 1).

y = sin(x² + 1)

1 The inside

There is something inside: x² + 1.

Call it u.

u = x² + 1

Then y is just sin u.

y = sin u

2 The outside

The derivative of sin x is cos x.

sin x → cos x

First the outside.

The rule on sin u gives cos u.

dydu = cos u

3 The inside's slope

Now the inside.

The power rule on x² + 1 gives 2x.

dudx = 2x

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

dydx = dydu × dudx

Put the inside back in place of u.

= cos(x² + 1) × 2x

Careful!

Do not stop here!

We must multiply by the inside's slope, 2x.

Careful: not cos(x² + 1) ✗

Write the numbers in front.

So dy/dx = 2x cos(x² + 1).

= 2x cos(x² + 1)

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives −0.8323, with a calculator in radians.

dy/dx at x = 1: −0.8323

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

A calculator gives about −0.83.

Very close to −0.8323.

Right!

y(1.001) − y(1)0.001 ≈ −0.83

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.

Answer