If z = x2 tan

tan

Prove that

a'z

=

дхду

o'z x2-y

дудх

=

x+

To prove the identities given, we'll need to clarify and carefully write down the mathematical expressions. Based on your text, it seems you're dealing with partial derivatives involving variables \( x \), \( y \), and a function \( z \). However, the notation appears incomplete or unclear. 

Let's try to parse the provided text and then proceed with the proof.

### Interpretation of Given Equations

1. **Original Equation**: 
   \[
   z = x^2 \tan(y)
   \]

2. **Partial Derivative Equation to Prove**:
   \[
   \frac{\partial z}{\partial x} = x^2 \tan(y)
   \]

   \[
   \frac{\partial z}{\partial y} = x^2 - y
   \]

Based on common notation and the equations written, it seems like we're looking to compute partial derivatives of \( z \) with respect to \( x \) and \( y \), but there are some inconsistencies with standard trigonometric and calculus rules. Let's go step by step to interpret correctly.

### Step-by-Step Derivation

Given:

\[
z = x^2 \tan(y)
\]

**Step 1: Compute \(\frac{\partial z}{\partial x}\)**

Given \( z = x^2 \tan(y) \), the partial derivative of \( z \) with respect to \( x \) treats \( y \) as a constant:

\[
\frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (x^2 \tan(y))
\]

Applying the derivative rule:

\[
\frac{\partial z}{\partial x} = 2x \tan(y)
\]

**Step 2: Compute \(\frac{\partial z}{\partial y}\)**

The partial derivative of \( z \) with respect to \( y \) treats \( x \) as a constant:

\[
\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (x^2 \tan(y))
\]

Applying the derivative rule:

\[
\frac{\partial z}{\partial y} = x^2 \sec^2(y)
\]

### Conclusion

Given the interpretation above, here are the correct expressions for the partial derivatives:

1. \[
\frac{\partial z}{\partial x} = 2x \tan(y)
\]

2. \[
\frac{\partial z}{\partial y} = x^2 \sec^2(y)
\]

It seems there might be a typo or error in the equations you were asked to prove. With the standard interpretation, these are the correct derivatives based on the function provided.

Would you like more details or have any questions about these steps?

### Related Questions

1. How do you find the second partial derivatives for the function \( z = x^2 \tan(y) \)?
2. What is the geometric interpretation of partial derivatives in multivariable calculus?
3. How do the partial derivatives change if the function \( z \) were \( z = x^2 + y^2 \)?
4. How do you compute the partial derivative if the function includes exponential functions, like \( z = e^{x^2 + y^2} \)?
5. What are some common applications of partial derivatives in physics and engineering?

**Tip:** When computing partial derivatives, always remember which variable is being treated as a constant. This helps avoid common mistakes, especially in functions involving multiple variables.

Learn how to prove the partial derivatives of the function z = x^2 tan(y) with respect to x and y. This problem involves applying the chain rule for partial derivatives and understanding trigonometric functions in multivariable calculus. Suitable for undergraduate students studying calculus.

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