Derivative of 4sin(5x)

Step by step

The question

Question: find dy/dx for y = 4 sin(5x).

1 The inside

There is something inside: 5x.

Call it u.

Then y is just 4 sin u.

2 The outside

The derivative of sin x is cos x.

First the outside.

The rule on 4 sin u gives 4 cos u.

3 The inside's slope

Now the inside.

The power rule on 5x gives 5.

4 Multiply

The chain rule: multiply the two.

dy/du times du/dx.

Put the inside back in place of u.

Careful!

Do not stop here!

We must multiply by the inside's slope, 5.

Write the numbers in front.

So dy/dx = 20 cos(5x).

5 Check

Let's check: the derivative is the slope.

At x = 1, the rule gives 5.6732, with a calculator in radians.

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

A calculator gives about 5.72.

Very close to 5.6732.

Right!

Let's look back.

First: the inside.

Then: the outside.

Then: the inside's slope.

Then: multiply.

Last: check.