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Partial Derivatives

April 10th, 2009 |

Author: Merrin

Up until now, we have dealt with the derivative of functions of a single variable. We can extend the notion of the derivative to functions of multiple variables. Partial derivatives are essential when extending calculus to multiple dimensions.

Definition 1. The partial derivative with respect to x of the function f(x,y) is given as

$\cfrac{\partial }{\partial x} f(x,y)=f_{x} (x,y)=\mathop{\lim }\limits_{\Delta x\to 0} \cfrac{f(x+\Delta x,y)-f(x,y)}{\Delta x}$

That is, treat the variable y as if it were a constant and differentiate the function as if it were f(x) where all previous differentiation rules apply.

Example 1. Find the partial derivative with respect to x of the following function

$f(x,y)=e^{xy} \sin (x^{2} +y)+\ln (x+y)$

Solution 1. Using our known differentiation rules and treating y as a constant we find.

$\cfrac{\partial }{\partial x} f(x,y)=ye^{xy} \sin (x^{2} +y)+e^{xy} \cos (x^{2} +y)2x+\cfrac{1}{x+y}$

As a bit of notation, we can also define the nth partial derivative with respect to x or y as

$\cfrac{\partial ^{n} }{\partial x^{n} } f(x,y)\quad \quad \quad \quad \quad \cfrac{\partial ^{n} }{\partial y^{n} } f(x,y)$

Theorem 1. If the mixed partial derivatives exist and are continuous at a point then they are equal. It doesn’t matter the order of taking the derivatives.

$\cfrac{\partial ^{2} }{\partial x\partial y} f(x,y)=\cfrac{\partial ^{2} }{\partial y\partial x} f(x,y)$

Theorem 2. Whereas for a function of a single variable linearization gives us

$df=f'(x)dx$

For a function of two variables linearization gives us

$df=\cfrac{\partial f}{\partial x} dx+\cfrac{\partial f}{\partial y} dy$

Another bit of notation. Sometimes partial derivatives are taken holding various variable constant. When it is necessary to explicitly state which variable is held constant, the variables held constant are written below the derivative.

$df=\left(\cfrac{\partial f}{\partial x} \right)_{y} dx+\left(\cfrac{\partial f}{\partial y} \right)_{x} dy$

Partial derivatives are useful when we want to calculate instantaneous rates of change, but we are dealing with multiple variables. Partial derivatives let us calculate the change of one variable out of many at a time. Partial derivatives are used extensively in many aspects of physics, engineering, and mathematics.

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