Partial Derivative & Gradient | Mathe by Daniel Jung

Summary

This video explains the transition from single-variable derivatives to multivariable calculus. It begins with a simple function f(x) = x² and its derivative, then expands to a function of two variables, f(x, y) = x² + y², illustrating how outputs form a 3D surface. The concept of partial derivatives is introduced, showing how to differentiate with respect to one variable while treating the other as a constant. Finally, the video defines the gradient as a vector containing all partial derivatives, highlighting its role in pointing toward the direction of steepest increase, making multivariable functions more intuitive and visually understandable.

Takeaways

• 😀 The example starts with a basic function f(x) = x², where the output is y, and the graph is in two dimensions (x, y).
• 😀 In the extended case, the function becomes f(x, y), where the output is z, moving the graph into three dimensions (x, y, z).
• 😀 The derivative of f(x) = x² is simple: f'(x) = 2x.
• 😀 When considering partial derivatives, the variable that is held constant (such as y) is treated as a constant during differentiation.
• 😀 The partial derivative of f with respect to x, treating y as constant, is 2x * y².
• 😀 In partial differentiation, treating y as a constant means that y² becomes a constant, and its derivative is zero.
• 😀 Similarly, the partial derivative of f with respect to y, treating x as constant, is 2y * x².
• 😀 Partial derivatives calculate how a multi-variable function changes with respect to one variable at a time, while others are held constant.
• 😀 The gradient is the combination of all partial derivatives, representing the rate of change of the function in all directions.
• 😀 In this example, the gradient is expressed as the vector [2x, 2y], combining the partial derivatives with respect to x and y.

Q & A

What is the purpose of partial derivatives in this example?

Partial derivatives are used to calculate the rate of change of a function with respect to one variable while holding the other variables constant. In the given example, the function f(x, y) is dependent on both x and y, so the partial derivatives with respect to each variable are calculated separately.

What is the function used in the example?

The function used in the example is f(x) = x², and it is later extended to f(x, y) = x² * y², which introduces a third dimension, with x and y as inputs and z as the output.

What happens when you switch from a two-dimensional function to a three-dimensional one?

When switching from a two-dimensional function, f(x), to a three-dimensional function, f(x, y), the function now depends on two variables, x and y, instead of just x. The output becomes a value z, representing a point in 3D space.

How is the partial derivative of f with respect to x calculated?

The partial derivative of f(x, y) = x² * y² with respect to x is calculated by treating y as a constant, which results in the derivative 2x * y².

How is the partial derivative of f with respect to y calculated?

The partial derivative of f(x, y) = x² * y² with respect to y is calculated by treating x as a constant. This gives the derivative 2y * x².

Why do we treat y as a constant when differentiating with respect to x?

When calculating a partial derivative with respect to x, we treat y as a constant because we are only interested in how the function changes with respect to x, holding y fixed.

What is the significance of the gradient in the context of partial derivatives?

The gradient is a vector that combines all partial derivatives of a function. It indicates the direction and rate of the steepest increase of the function, providing valuable information about the function’s behavior in multivariable space.

What does the gradient represent in this example?

In this example, the gradient represents the combined partial derivatives of the function f(x, y) = x² * y², which are 2x * y² and 2y * x². These derivatives describe the rate of change of the function in the x and y directions.

What would the gradient look like for a different function, such as f(x, y) = x³ * y³?

For the function f(x, y) = x³ * y³, the partial derivative with respect to x would be 3x² * y³, and the partial derivative with respect to y would be 3x³ * y². The gradient would then be the vector (3x² * y³, 3x³ * y²).

What is the difference between a partial derivative and a regular derivative?

A regular derivative measures the rate of change of a function with respect to one variable, while a partial derivative measures the rate of change with respect to one variable, holding all other variables constant. Partial derivatives are used for functions of multiple variables.