# Partial Derivatives and the Gradient of a Function

TLDRThis lesson reintroduces differentiation notation using d/dx and del operators to represent taking derivatives and gradients. It explains how to take partial derivatives of multivariable functions to find rates of change in specific directions. The gradient vector combines these partial derivatives. An example shows finding a three-dimensional function's gradient by taking its three partial derivatives. The lesson concludes by noting del and gradients have additional applications that will be covered further.

###### Takeaways

- π Revisiting differentiation and introducing new notation d/dx as a differential operator that represents taking the derivative
- π― The differential operator d/dx distributes across sums and differences of functions
- π§ Partial derivatives βf/βx allow taking derivatives of multivariable functions with respect to one variable at a time
- π The gradient combines partial derivatives into a vector showing rates of change
- π Del (β) represents a vector of partial derivative operators that can be used to take the gradient
- π‘ The gradient vector points in the direction of maximum change and has a magnitude equal to the maximum rate of change
- π Partial derivatives are taken by treating other variables as constants and differentiating normally
- π’ The chain rule still applies when taking partial derivatives
- π€ Variables like x, y and z can be inside of vectors when working with vector fields
- π€― Del has additional uses beyond gradients that will be covered later

###### Q & A

### What is the new notation introduced for taking derivatives instead of using f prime?

-The new notation is d/dx or DF/DX, which represents a differential operator that acts on the function.

### How can we interpret the rate of change for a function with multiple variables like f(x,y)?

-We take partial derivatives, like βf/βx and βf/βy, to find the rates of change in specific directions independently.

### What is the gradient of a multivariable function?

-The gradient is a vector made up of all the function's partial derivatives, giving the direction of maximum change.

### What does the del symbol β represent?

-Del (β) represents a vector of all the partial derivative operators, so βf gives the gradient of f.

### How do we calculate partial derivatives?

-Treat the variable you are differentiating with respect to as normal, and treat all other variables as constants.

### What is an example use of the product rule with the new notation?

-d/dx(fg) = f dg/dx + g df/dx

### What is an example use of the chain rule with the new notation?

-d/dx(f(g(x))) = f'(g(x)) g'(x)

### Why are we using variables like x and y inside vectors?

-We are representing partial derivatives and gradients as vectors. This will be discussed more when we cover vector fields.

### What are some other uses of the del operator?

-Del has various uses besides gradients, like in curl, divergence, Laplacian, and integration.

### How can the gradient vector be interpreted geometrically?

-The gradient points in the direction of maximum increase of the function and has a magnitude equal to the maximum rate of change.

###### Outlines

##### π Introducing Partial Derivatives and the Gradient Vector

This paragraph introduces partial derivatives as a way to find the rate of change of a function with multiple variables in specific directions independently. It explains how to take partial derivatives by treating the other variables as constants. It then introduces the gradient vector, which combines all the partial derivatives of a function into a single vector that points in the direction of maximum change.

##### π Applying Partial Derivatives and the Gradient

This paragraph provides examples of finding partial derivatives and gradients for functions with multiple variables. It shows how to take the partial derivatives by treating other variables as constants and following derivative rules. It explains how to construct the gradient vector from the partial derivatives. An example function is shown and its gradient is derived step-by-step.

###### Mindmap

###### Keywords

##### π‘Differentiation

##### π‘Derivative

##### π‘Differential operator

##### π‘Partial derivative

##### π‘Gradient

##### π‘Vector

##### π‘Rate of change

##### π‘Function

##### π‘Power rule

##### π‘Chain rule

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###### Transcripts

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