https://file.aiquickdraw.com/mathbot/file/1726732837511i90wosss.jpg

We are tasked with finding \( \frac{dz}{dt} \) for the function:

\[
z = f(x, y) = x^2 + y^3
\]

where \( x = t^2 \) and \( y = 2t \).

### Solution:
To find \( \frac{dz}{dt} \), we need to apply the chain rule. The chain rule for a function \( z = f(x, y) \), where both \( x \) and \( y \) are functions of \( t \), is given by:

\[
\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}
\]

Now, let’s compute each term step by step:

1. **Partial derivative of \( z \) with respect to \( x \):**
   \[
   \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^3) = 2x
   \]

2. **Partial derivative of \( z \) with respect to \( y \):**
   \[
   \frac{\partial z}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^3) = 3y^2
   \]

3. **Derivative of \( x = t^2 \) with respect to \( t \):**
   \[
   \frac{dx}{dt} = \frac{d}{dt}(t^2) = 2t
   \]

4. **Derivative of \( y = 2t \) with respect to \( t \):**
   \[
   \frac{dy}{dt} = \frac{d}{dt}(2t) = 2
   \]

Now, applying the chain rule:

\[
\frac{dz}{dt} = \left( 2x \right) \cdot \frac{dx}{dt} + \left( 3y^2 \right) \cdot \frac{dy}{dt}
\]

Substitute \( x = t^2 \) and \( y = 2t \) into the equation:

\[
\frac{dz}{dt} = 2(t^2) \cdot (2t) + 3(2t)^2 \cdot 2
\]

Simplify each term:

\[
\frac{dz}{dt} = 2t^2 \cdot 2t + 3 \cdot 4t^2 \cdot 2
\]

\[
\frac{dz}{dt} = 4t^3 + 24t^2
\]

Thus, the derivative is:

\[
\frac{dz}{dt} = 4t^3 + 24t^2
\]

Would you like further details or have any questions?

---

Here are five related questions to deepen your understanding:

1. How would the chain rule change if \( z \) had more variables, such as \( z = f(x, y, w) \)?
2. What is the interpretation of partial derivatives in multivariable functions?
3. How does implicit differentiation apply to related rates problems?
4. Can you apply this process to different functions of \( x \) and \( y \) with respect to \( t \)?
5. What if \( x \) and \( y \) were functions of two variables, say \( t \) and \( u \)?

**Tip:** Always apply the chain rule carefully by ensuring that each term is properly differentiated with respect to the desired variable.

Find dz/dt for the following functions: z = f(x, y) = x^2 + y^3 with x = t^2 and y = 2t.

Explore the application of the chain rule in calculus to calculate the time derivative dz/dt of the function z = x^2 + y^3, where x = t^2 and y = 2t. This detailed guide covers partial derivatives and their role in complex variable dependency.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation