Calculate The Following Derivative If Y Cos 4x9

Introduction

When you need to find the derivative of a product of functions like y cos 4x9, you'll need to use the product rule. This is a fundamental concept in calculus that helps us find the rate of change of functions that are multiplied together.

The product rule states that if you have two functions u(x) and v(x), then the derivative of their product is:

In our case, u = y and v = cos 4x9. We'll need to find the derivatives of both functions separately before applying the product rule.

Formula

The derivative of y cos 4x9 can be found using the product rule:

We'll need to know the derivative of y and the derivative of cos 4x9 separately. The derivative of cos 4x9 can be found using the chain rule.

Step-by-Step Calculation

1. Identify the two functions: u = y and v = cos 4x9
2. Find the derivative of u with respect to x: dy/dx
3. Find the derivative of v with respect to x using the chain rule:
   1. Let w = 4x9 → dw/dx = 36
   2. Let z = cos w → dz/dw = -sin w
   3. By chain rule: dz/dx = dz/dw × dw/dx = -sin(4x9) × 36
4. Apply the product rule:

   d/dx [y cos 4x9] = (dy/dx) cos 4x9 + y (-36 sin 4x9)

Worked Example

Let's say y = 5x² + 3x + 7. We'll calculate the derivative of y cos 4x9.

1. First find dy/dx:

   dy/dx = 10x + 3

2. We already found d/dx [cos 4x9] = -36 sin 4x9
3. Now apply the product rule:

   d/dx [y cos 4x9] = (10x + 3) cos 4x9 + (5x² + 3x + 7)(-36 sin 4x9)

This gives us the complete derivative expression for this specific case.