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The problem asks to find the equation of the tangent line to the curve \(\cos(x - y) = -3x - 3y\) at the point \(\left(\frac{5\pi}{4}, -\frac{5\pi}{4}\right)\).

### Steps:
1. **Implicit Differentiation:**
   We begin by differentiating both sides of \(\cos(x - y) = -3x - 3y\) with respect to \(x\), applying the chain rule and treating \(y\) as a function of \(x\).
   \[
   \frac{d}{dx}\left(\cos(x - y)\right) = \frac{d}{dx}(-3x - 3y)
   \]
   Using the chain rule:
   \[
   -\sin(x - y) \cdot \left(1 - \frac{dy}{dx}\right) = -3 - 3\frac{dy}{dx}
   \]
   Simplifying the equation:
   \[
   -\sin(x - y) + \sin(x - y)\frac{dy}{dx} = -3 - 3\frac{dy}{dx}
   \]
   Rearranging to collect terms involving \(\frac{dy}{dx}\):
   \[
   \sin(x - y)\frac{dy}{dx} + 3\frac{dy}{dx} = \sin(x - y) - 3
   \]
   Factoring out \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx}(\sin(x - y) + 3) = \sin(x - y) - 3
   \]
   Solving for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = \frac{\sin(x - y) - 3}{\sin(x - y) + 3}
   \]

2. **Substitute the Point \(\left(\frac{5\pi}{4}, -\frac{5\pi}{4}\right)\):**
   Now, substitute \(x = \frac{5\pi}{4}\) and \(y = -\frac{5\pi}{4}\) into the expression for \(\frac{dy}{dx}\). First, calculate \(\sin\left(x - y\right)\):
   \[
   x - y = \frac{5\pi}{4} - \left(-\frac{5\pi}{4}\right) = \frac{5\pi}{4} + \frac{5\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2}
   \]
   Since \(\sin\left(\frac{5\pi}{2}\right) = 1\), we have:
   \[
   \frac{dy}{dx} = \frac{1 - 3}{1 + 3} = \frac{-2}{4} = -\frac{1}{2}
   \]

3. **Equation of the Tangent Line:**
   The slope of the tangent line is \(\frac{dy}{dx} = -\frac{1}{2}\). The point of tangency is \(\left(\frac{5\pi}{4}, -\frac{5\pi}{4}\right)\). The equation of the tangent line is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting the point and slope:
   \[
   y + \frac{5\pi}{4} = -\frac{1}{2}\left(x - \frac{5\pi}{4}\right)
   \]
   Simplifying:
   \[
   y + \frac{5\pi}{4} = -\frac{1}{2}x + \frac{5\pi}{8}
   \]
   \[
   y = -\frac{1}{2}x + \frac{5\pi}{8} - \frac{5\pi}{4}
   \]
   \[
   y = -\frac{1}{2}x - \frac{5\pi}{8}
   \]

So, the equation of the tangent line is:
\[
y = -\frac{1}{2}x - \frac{5\pi}{8}
\]

Would you like any further clarification?

Here are 5 related questions to expand your understanding:
1. How is implicit differentiation different from regular differentiation?
2. What is the geometric significance of the derivative at a point?
3. How do we interpret the slope of a tangent line in terms of the behavior of the function?
4. What are other ways to find tangent lines for parametric or polar curves?
5. How does implicit differentiation work when there are more than two variables?

**Tip:** When applying implicit differentiation, always treat \(y\) as a function of \(x\) and apply the chain rule carefully.

Bestäm en ekvation för tangenten till kurvan cos(x - y) = -3x - 3y i punkten (5/4 π, -5/4 π).

Learn how to find the tangent line to the curve cos(x - y) = -3x - 3y at the point (5π/4, -5π/4). This problem involves applying implicit differentiation and understanding the relationship between x and y in the trigonometric equation.

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