Calculate The Following Derivative If Y Cos 4x 9

This calculator helps you find the derivative of the function y = cos(4x) + 9. The derivative represents the rate of change of the function with respect to x. Understanding derivatives is essential in calculus, physics, and engineering.

How to Calculate the Derivative

The derivative of a function y = f(x) is denoted as dy/dx or f'(x). For the function y = cos(4x) + 9, we'll use the chain rule to find its derivative.

Derivative Formula

If y = cos(u) + C, where C is a constant, then dy/dx = -sin(u) * du/dx.

In our case, u = 4x, so du/dx = 4.

The derivative of a constant is always zero. Therefore, the derivative of the +9 term is 0.

Step-by-Step Solution

Identify the function: y = cos(4x) + 9
Apply the chain rule to the cos(4x) term:
- Let u = 4x
- dy/dx = -sin(u) * du/dx
- du/dx = 4 (since the derivative of 4x is 4)
Combine the results:
- dy/dx = -sin(4x) * 4 + 0 = -4sin(4x)

Worked Example

Let's find the derivative of y = cos(4x) + 9 at x = π/8.

Example Calculation

1. First, find the derivative: dy/dx = -4sin(4x)

2. Substitute x = π/8: dy/dx = -4sin(4 * π/8) = -4sin(π/2)

3. Since sin(π/2) = 1, the result is -4 * 1 = -4

The derivative at x = π/8 is -4, which means the rate of change of the function at that point is -4 units per unit change in x.

Frequently Asked Questions

What is the derivative of cos(4x)?: The derivative of cos(4x) is -4sin(4x). This comes from applying the chain rule to the cosine function.
Why is the derivative of a constant zero?: The derivative of any constant term (like +9 in our function) is always zero because constants do not change as x changes.
How do I use the chain rule for derivatives?: The chain rule states that if you have a composite function like cos(4x), you first find the derivative of the outer function (cosine) and then multiply it by the derivative of the inner function (4x).
What does a negative derivative mean?: A negative derivative indicates that the function is decreasing at that point. In our example, the function decreases at a rate of 4 units per unit change in x.