Question

Find the slope of the tangent line of cos(3x) at x=2.

Answer 1

Main Answer:

The slope of the tangent line to the function
`\(y = \cos(3x)\) at \(x = 2\)` can be found by taking the derivative of the function with respect to
`\(x\)` and then evaluating it at
`\(x = 2\)`.

Step-by-step explanation:

1. **Find the Derivative:**

Start by finding the derivative of the given function
`\(y = \cos(3x)\)` with respect to
`\(x\)`. Use the chain rule since there is a composition of functions. The derivative of
`\(\cos(u)\)` with respect to
`\(u\) is \(-\sin(u)\)`, and then multiply by the derivative of the inner function.

`\[ (dy)/(dx) = -\sin(3x) \cdot (d(3x))/(dx) \]`

`\[ (dy)/(dx) = -3\sin(3x) \]`

2. **Evaluate at
`\(x = 2\)`:**

Substitute
`\(x = 2\)` into the derivative to find the slope at that specific point.

`\[ \text{Slope at } x = 2: \quad m = -3\sin(6) \]`

3. **Calculate the Numerical Value:**

Use a calculator to find the numerical value of
`\(-3\sin(6)\)`.

So, the slope of the tangent line to
`\(y = \cos(3x)\) at \(x = 2\) is \(-3\sin(6)\)`. You can use a calculator to get the approximate numerical value if needed.