This document presents the partial derivatives of an exponential function with two variables.
The function has the form \( f(x, y) = e^{3x^2y} \), and we will examine its first and second order partial derivatives.
Original Function
This is an exponential function where the exponent is a function of both x and y.
First Order Partial Derivatives
Partial derivative with respect to x. Apply chain rule: derivative of \(e^u\) is \(e^u \cdot u_x\), where \(u = 3x^2y\) and \(u_x = 6xy\).
Partial derivative with respect to y. Apply chain rule: derivative of \(e^u\) is \(e^u \cdot u_y\), where \(u = 3x^2y\) and \(u_y = 3x^2\).
Second Order Partial Derivatives
Second partial derivative with respect to x. Differentiate \(\frac{\partial f}{\partial x}\) with respect to x using product rule and chain rule.
Second partial derivative with respect to y. Differentiate \(\frac{\partial f}{\partial y}\) with respect to y using chain rule.