Background
- (Ordinary) derivatives
- Multivariable functions
- Graphs of multivariable function
What we're building to
- For a multivariable function, like
, computing partial derivatives looks something like this:
This swirly-d symbol,
, often called "del", is used to distinguish partial derivatives from ordinary single-variable derivatives. Or, should I say ... to differentiate them.The reason for a new type of derivative is that when the input of a function is made up of multiple variables, we want to see how the function changes as we let just one of those variables change while holding all the others constant.
With respect to three-dimensional graphs, you can picture the partial derivative
by slicing the graph of with a plane representing a constant -value and measuring the slope of the resulting curve along the cut.
What is a partial derivative?
We'll assume you are familiar with the ordinary derivative
Interpret
as "a very tiny change in ".Interpret
as "a very tiny change in the output of ", where it is understood that this tiny change is whatever results from the tiny change to the input.
In fact, I think this intuitive feel for the symbol
For example, when you apply it to the graph of
How does this work for multivariable functions?
Consider some function with a two-dimensional input and a one-dimensional output.
There's nothing stopping us from writing the same expression,
can still represent a tiny change in the variable , which is now just one component of our input. can still represent the resulting change to the output of the function .
However, this ignores the fact that there is another input variable
Neither one of these derivatives tells the full story of how our function
You read the symbol
Example: Computing a partial derivative
Consider this function:
Suppose I asked you to evaluate
"What? But I haven't learned how yet!"
Don't worry, it's mostly just the same mechanics as an ordinary derivative.
From the introduction above, you should know that this is asking about the rate at which the output of
Since we only care about movement in the
Now, asking how
Concept check
What is the derivative of this function
Without pre-evaluating
Now suppose I asked you to find
You can start the same way, treating the
Or rather, since to emphasize that this is a multivariable function, we use the symbol
As a sanity check, you can plug in
"So, what's the difference between
Honestly, as far as I'm concerned, there's not really a difference between these operations. You could be pedantic and say one is only defined for single variable functions. But as far as intuition and computation go, they are one and the same, and the difference is just meant to clarify what type of function is being differentiated.
Interpreting partial derivatives with graphs
Consider this function:
Here is a video showing its graph rotating, just to get a feel for the three-dimensional nature of it.
Khan Academy video wrapper
See video transcript
Think about the partial derivative of
In terms of the graph, what does the value of this expression tell us about the behavior of the function
Treat as constant slice graph with plane
The first step when computing this value is to treat
This plane
What about
Reflection Question
In the picture to the right, the "curve" where the graph of
Is it really a line?
The partial derivative
Graphically, this means as we choose different values of
Khan Academy video wrapper
See video transcript
Phrasing and notation
Here are some of the phrases you might hear in reference to this
- "The partial derivative of
with respect to " - "Del f, del x"
- "Partial f, partial x"
- "The partial derivative (of
) in the -direction"
Alternate notation
In the same way that people sometimes prefer to write
A note about "del"
While it's common to refer to the partial symbol
A more formal definition
Although thinking of
The point of calculus is that we don't use any one tiny number, but instead consider all possible values and analyze what tends to happen as they approach a limiting value. The single variable derivative, for example, is defined like this:
represents the "tiny value" that we intuitively think of as .The
under the limit indicates that we care about very small values of , those approaching . is the change in the output that results from adding to the input, which is what we think of as .
Formally defining the partial derivative looks almost identical. If
Similarly, here's how the partial derivative with respect to
The point is that
People will often refer to this as the limit definition of a partial derivative.
Reflection question: How can we think about this limit definition in the context of the graphical interpretation above? What is
Summary
- For a multivariable function, like
, computing partial derivatives looks something like this:
This swirly-d symbol
, often called "del", is used to distinguish partial derivatives from ordinary single-variable derivatives.The reason for a new type of derivative is that when the input of a function is made up of multiple variables, we want to see how the function changes as we let just one of those variables change while holding all the others constant.
With respect to three-dimensional graphs, you can picture the partial derivative
by slicing the graph of with a plane representing a constant -value, and measuring the slope of the resulting cut.