Partial Derivatives

Front

What is a partial derivative?

Back

A partial derivative is the derivative of a function with respect to one variable while keeping the other variables constant.

Front

How do you denote the partial derivative of a function f with respect to x?

Back

The partial derivative of a function f with respect to x is denoted as f_x or ∂f/∂x.

Front

What is the formula for finding the partial derivative of a function f(x, y) = ax^n + by^m with respect to x?

Back

The partial derivative with respect to x is f_x = n * ax^(n-1).

Front

What is the formula for finding the partial derivative of a function f(x, y) = ax^n + by^m with respect to y?

Back

The partial derivative with respect to y is f_y = m * by^(m-1).

Front

Given f(x, y) = 5x^2 - 3xy + y^2, what is f_y?

Back

f_y = -3x + 2y.

Front

Given f(x, y) = x^2 + 3y, what is f_x?

Back

f_x = 2x.

Front

Given f(x, y) = x^2y + y^2x, what is f_y?

Back

f_y = x^2 + 2xy.

Front

For f(x, y) = ln(x + y), how do you find f_x?

Back

f_x = 1/(x + y).

Front

For f(x, y) = x/y, what is f_x?

Back

f_x = 1/y.

Front

What is the significance of mixed partial derivatives?

Back

Mixed partial derivatives are derivatives taken with respect to more than one variable, showing how a function changes as multiple variables change.