Differentiate (x(t)-3)^2 - (y(t)-2)^2 with respect to t, suppose x(t) = √2 cos(t) − √2/2 sin(t) + 3 and y(t) = √2 cos(t) + √2/2 cos(t)

To differentiate the expression \((x(t) - 3)^2 - (y(t) - 2)^2\) with respect to \(t\), we will use the chain rule.

### Step 1: Define the expression
\[
f(t) = (x(t) - 3)^2 - (y(t) - 2)^2
\]
where
\[
x(t) = \sqrt{2} \cos(t) - \frac{\sqrt{2}}{2} \sin(t) + 3
\]
and
\[
y(t) = \sqrt{2} \cos(t) + \frac{\sqrt{2}}{2} \cos(t)
\]

### Step 2: Differentiate the first term \((x(t) - 3)^2\)
Using the chain rule, the derivative of \((x(t) - 3)^2\) with respect to \(t\) is:
\[
\frac{d}{dt} (x(t) - 3)^2 = 2(x(t) - 3) \cdot \frac{dx(t)}{dt}
\]
Now, differentiate \(x(t)\):
\[
x(t) = \sqrt{2} \cos(t) - \frac{\sqrt{2}}{2} \sin(t) + 3
\]
\[
\frac{dx(t)}{dt} = -\sqrt{2} \sin(t) - \frac{\sqrt{2}}{2} \cos(t)
\]

Thus, the derivative of \((x(t) - 3)^2\) is:
\[
\frac{d}{dt} (x(t) - 3)^2 = 2(x(t) - 3) \left(-\sqrt{2} \sin(t) - \frac{\sqrt{2}}{2} \cos(t)\right)
\]

### Step 3: Differentiate the second term \((y(t) - 2)^2\)
Similarly, for \((y(t) - 2)^2\), we apply the chain rule:
\[
\frac{d}{dt} (y(t) - 2)^2 = 2(y(t) - 2) \cdot \frac{dy(t)}{dt}
\]
Now differentiate \(y(t)\):
\[
y(t) = \sqrt{2} \cos(t) + \frac{\sqrt{2}}{2} \cos(t)
\]
\[
\frac{dy(t)}{dt} = -\sqrt{2} \sin(t) - \frac{\sqrt{2}}{2} \sin(t)
\]

Thus, the derivative of \((y(t) - 2)^2\) is:
\[
\frac{d}{dt} (y(t) - 2)^2 = 2(y(t) - 2) \left(-\sqrt{2} \sin(t) - \frac{\sqrt{2}}{2} \sin(t)\right)
\]

### Step 4: Combine the results
Now subtract the two terms to get the derivative of the entire expression:
\[
\frac{d}{dt} f(t) = 2(x(t) - 3) \left(-\sqrt{2} \sin(t) - \frac{\sqrt{2}}{2} \cos(t)\right) - 2(y(t) - 2) \left(-\sqrt{2} \sin(t) - \frac{\sqrt{2}}{2} \sin(t)\right)
\]
You can simplify this further by substituting the expressions for \(x(t)\) and \(y(t)\), but this is the general derivative with respect to \(t\).

Would you like the result simplified, or do you have any other questions?

### Related questions:
1. What is the physical interpretation of \(x(t)\) and \(y(t)\) in this context?
2. How would you compute the second derivative of this expression?
3. How would you handle differentiation if the expressions for \(x(t)\) and \(y(t)\) were parametric functions in 3D?
4. What happens if \(x(t)\) or \(y(t)\) had discontinuities?
5. How does the chain rule apply to more complex nested functions?

### Tip:
Always check if your functions simplify before differentiating—they often reveal shortcuts!

Learn how to differentiate the expression (x(t)-3)^2 - (y(t)-2)^2 with respect to t using the chain rule. This problem involves parametric functions, calculus concepts, and the application of trigonometric derivatives. Suitable for students at the undergraduate level studying Calculus I or II.

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