Given f, of, x, equals, 3, cosine, left parenthesis, x, right parenthesis, commaf(x)=3cos(x), write the equation of the line tangent to y, equals, f, of, xy=f(x) when x, equals, start fraction, 3, pi, divided by, 4, end fraction, .x= 
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3π
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 .

To find the equation of the tangent line to \( y = f(x) = 3\cos(x) \) at \( x = \frac{3\pi}{4} \), we need two things: the slope of the tangent line and the point of tangency.

### Step 1: Find the derivative of \( f(x) \)
The derivative of \( f(x) = 3\cos(x) \) gives the slope of the tangent line. Using the derivative rule for \( \cos(x) \), we have:

\[
f'(x) = -3\sin(x)
\]

### Step 2: Evaluate the derivative at \( x = \frac{3\pi}{4} \)
To find the slope of the tangent line, substitute \( x = \frac{3\pi}{4} \) into the derivative:

\[
f'\left(\frac{3\pi}{4}\right) = -3\sin\left(\frac{3\pi}{4}\right)
\]

Now, \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \), so:

\[
f'\left(\frac{3\pi}{4}\right) = -3 \times \frac{\sqrt{2}}{2} = -\frac{3\sqrt{2}}{2}
\]

Thus, the slope of the tangent line is \( m = -\frac{3\sqrt{2}}{2} \).

### Step 3: Find the point of tangency
We now calculate \( f\left(\frac{3\pi}{4}\right) \):

\[
f\left(\frac{3\pi}{4}\right) = 3\cos\left(\frac{3\pi}{4}\right)
\]

Since \( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), we get:

\[
f\left(\frac{3\pi}{4}\right) = 3 \times \left(-\frac{\sqrt{2}}{2}\right) = -\frac{3\sqrt{2}}{2}
\]

So, the point of tangency is \( \left( \frac{3\pi}{4}, -\frac{3\sqrt{2}}{2} \right) \).

### Step 4: Write the equation of the tangent line
The equation of the tangent line is given by the point-slope form:

\[
y - y_1 = m(x - x_1)
\]

Substitute \( m = -\frac{3\sqrt{2}}{2} \), \( x_1 = \frac{3\pi}{4} \), and \( y_1 = -\frac{3\sqrt{2}}{2} \):

\[
y + \frac{3\sqrt{2}}{2} = -\frac{3\sqrt{2}}{2} \left( x - \frac{3\pi}{4} \right)
\]

Simplifying this equation gives the tangent line equation:

\[
y = -\frac{3\sqrt{2}}{2}x + \frac{9\pi\sqrt{2}}{8}
\]

Thus, the equation of the tangent line is:

\[
y = -\frac{3\sqrt{2}}{2}x + \frac{9\pi\sqrt{2}}{8}
\]

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Would you like further clarification on any steps or more details?

Here are five related questions for practice:
1. What is the derivative of \( f(x) = 5\sin(x) \)?
2. Find the equation of the tangent line to \( f(x) = \sin(x) \) at \( x = \frac{\pi}{6} \).
3. What is the slope of the tangent line to \( f(x) = \cos(x) \) at \( x = 0 \)?
4. How do you find the second derivative of \( f(x) = 3\cos(x) \)?
5. Determine the points where the tangent line to \( f(x) = \cos(x) \) is horizontal.

**Tip**: The derivative of a function at a point gives the slope of the tangent line at that point.

Given f(x) = 3cos(x), write the equation of the line tangent to y = f(x) when x = 3π/4.

Learn how to find the equation of the tangent line to the function f(x) = 3cos(x) at x = 3π/4. This problem covers key calculus concepts, including differentiation and trigonometric functions, and walks through each step to solve the equation using the point-slope form of a line.

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