Find du if u = x² + y² + 2², where n = e ^ t dt y = e fint, z = e ^ t Cast

Let's first organize and interpret the information in the problem:

Given:
- \( u = x^2 + y^2 + 2^2 \)
- \( n = e^t \, dt \)
- \( y = e^{\text{fint}} \)
- \( z = e^t \, \text{Cast} \)

### Step 1: Clarifying the notation and problem
There might be some notation issues. For example, it's unclear what "fint" and "Cast" refer to. If "fint" refers to an indefinite integral (or antiderivative), and "Cast" represents some constant, I will assume the following:
- \( n = \int e^t \, dt = e^t + C_1 \) (integrating \( e^t \) with respect to \( t \)).
- \( y = e^{F(t)} \), where \( F(t) \) is an antiderivative of some function \( f(t) \) (assuming "fint" means an integral with respect to \( t \)).
- \( z = e^t + C_2 \) (assuming "Cast" is a constant).

Now, we need to find \( du \), the differential of \( u \).

### Step 2: Differentiating \( u = x^2 + y^2 + 2^2 \)

#### 1. Write \( u \):
\[
u = x^2 + y^2 + 4
\]

#### 2. Differentiate \( u \) with respect to \( t \):

Using the chain rule, we get:
\[
du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy + \frac{\partial u}{\partial t} dt
\]

- \( \frac{\partial u}{\partial x} = 2x \), so \( 2x \, dx \).
- \( \frac{\partial u}{\partial y} = 2y \), so \( 2y \, dy \).
- Since \( u \) does not explicitly depend on \( t \), \( \frac{\partial u}{\partial t} = 0 \).

Now, using the chain rule on \( y = e^{F(t)} \), we know:
\[
dy = \frac{d}{dt}(e^{F(t)}) dt = e^{F(t)} F'(t) dt
\]

#### 3. Express \( du \):
\[
du = 2x \, dx + 2y \, e^{F(t)} F'(t) \, dt
\]

### Step 3: Conclusion
So, the differential \( du \) is:

\[
du = 2x \, dx + 2y \, e^{F(t)} F'(t) \, dt
\]

Let me know if you'd like more details, or if there's any clarification needed!

### Expanding the concepts:
Here are 5 related questions that build upon this:
1. How do you differentiate implicitly if \( x \) and \( y \) depend on \( t \)?
2. What are some methods for finding antiderivatives of exponential functions?
3. How do constants impact the integration of functions?
4. How does the chain rule apply to multivariable functions?
5. How do you compute the total derivative of a function with multiple variables?

### Tip:
When differentiating functions involving multiple variables, always check if each variable is dependent on the one you're differentiating with respect to, as that may require the chain rule.

Explore how to differentiate u = x² + y² + 4 with respect to multiple variables using the chain rule. This problem involves differentiating exponential functions, applying the chain rule, and working with multivariable calculus concepts. Suitable for undergraduate calculus students.

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