Question

Let \( a, b \in \mathbb{R} \) and let \( f: \mathbb{R} \to \mathbb{R} \) be a thrice differentiable function. If \( z = e^u f(v) \), where \( u = ax + by \) and \( v = ax - by \), then which one of the following is TRUE?

• A. \( b^2 z_{xx} - a^2 z_{yy} = 4a^2 b^2 e^u f'(v) \)
• B. \( b^2 z_{xx} - a^2 z_{yy} = -4e^u f'(v) \)
• C. \( b z_x + a z_y = abz \)
• D. \( b z_{xx} + a z_y = -abz \)

Hint:

When differentiating composite functions like \( z = e^u f(v) \), carefully apply the chain rule and watch out for mixed derivatives.

Answer 1

The Correct Option is A

Step 1: Understanding the variables.
We are given the function \( z = e^u f(v) \), where \( u = ax + by \) and \( v = ax - by \). To find the second partial derivatives of \( z \), we need to apply the chain rule.

Step 2: Differentiating the expression for \( z \).
First, we differentiate \( z = e^u f(v) \) with respect to \( x \) and \( y \), using the chain rule to find \( z_x \) and \( z_y \), and then compute the second derivatives \( z_{xx} \) and \( z_{yy} \). The final result is the equation given in option (A).

Step 3: Conclusion.
The correct answer is (A) \( b^2 z_{xx} - a^2 z_{yy} = 4a^2 b^2 e^u f'(v) \).