23 Lesson Partial Derivatives When we are asked to find the derivative of a function of a single variable, f ( x), we know exactly what to do. However, when we have a function of two variables, there is some ambiguity. With a function of two variables, we can find the slope of the tangent line at a point P from an infinite number of directions. We will only consider two directions, either parallel to the x axis or parallel to the y axis. When we do this, we fix one of the variables. Then we can find the derivative with respect to the other variable.
So, if we fix y, we can find the derivative of the function with respect to the variable x. And if we fix x, we can find the derivative of the function with respect to the variable y. These derivatives are called partial derivatives.
First Partial Derivatives
We will use two different notations:
f or f x consider y as a constant x
f or f y consider x as a constant y
We can also evaluate a partial derivative at a given point. Example 1: Find the first partial derivatives of the function f ( x, y ) = x 2 3xy 2 + 4 y 2 . Then evaluate the first partials at (-1, 2).
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Example 2: Evaluate the first partial derivatives of f ( x) = 2 x 2 y 2 5 ye x at the point (1, 2).
Example 3: Find the first partial derivatives the of function f ( x, y ) =
xy . x + y2
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Example 4: Find the first partial derivatives of the function f ( x, y ) = y 1 x 2 .
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Example 5: Find the first partial derivatives of the function f ( x, y ) = (x 2 2 xy + 3 y 2 ) .
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Example 6: Find the first partial derivatives of the function f ( x, y ) = ln 3 x 2 4 y 2
(
)
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Second-Order Partial Derivatives Sometimes we will need to find the second-order partial derivatives. To find a second-order partial derivative, you will take respective partial derivatives of the first partial derivative. There are a total of 4 second-order partial derivatives. There are two notations, but we will only use one of them.
f xx =
2 f f = 2 x x x
2 f f = 2 y y y
f xy =
2 f f = xy y x
2 f f = yx x y
f yy =
f yx =
Example 7: Find the second-order partial derivatives of the function f ( x, y ) = 3x 2 y 2 5 x 2 + 10 y .
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Example 8: Find the second-order partial derivatives of the function f ( x, y ) = 3x 2 x 3 y 3 + 5 xy + 6 y 3 .
From this section, you should be able to Find first order partial derivatives Find second order partial derivatives Evaluate partial derivatives at a given point
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