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Partial DerivativesPartial Derivatives
GoalsGoals Define Define partial derivativespartial derivatives Learn Learn notationnotation and and rules for calculatingrules for calculating
If If ff is a function of two variables is a function of two variables xx and and yy, , suppose we let only suppose we let only xx vary while keeping vary while keeping yy fixed, say fixed, say yy = = bb, where , where bb is a constant. is a constant.
Then we are really considering a function Then we are really considering a function of a single variable of a single variable xx, namely, , namely, gg((xx) = ) = ff((xx, , bb).).
If If gg has a derivative at has a derivative at aa, then we call it , then we call it the the partial derivative of f with respect to partial derivative of f with respect to x at x at ((aa, , bb) and denote it by ) and denote it by ffxx((aa, , bb).).
Introduction (cont’d)Introduction (cont’d)
ThusThus
The definition of a derivative givesThe definition of a derivative gives
Introduction (cont’d)Introduction (cont’d)
Similarly, the Similarly, the partial derivative of f partial derivative of f with respect to y at with respect to y at ((aa, , bb), denoted ), denoted by by ffyy((aa, , bb), is obtained by holding ), is obtained by holding xx = =
aa and finding the ordinary derivative and finding the ordinary derivative at at bb of the function of the function GG((yy) = ) = ff((aa, , yy):):
DefinitionDefinition
If we now let the point (If we now let the point (aa, , bb) vary, ) vary, ffxx
and and ffyy become functions of two become functions of two
variables:variables:
NotationsNotations
There are many alternate notations There are many alternate notations for partial derivatives:for partial derivatives:
To give a geometric interpretation of To give a geometric interpretation of partial derivatives, we recall that the partial derivatives, we recall that the equation equation zz = = ff((xx, , yy) represents a surface ) represents a surface SS..
If If ff((aa, , bb) = ) = cc, then the point , then the point PP((aa, , bb, , cc) ) lies on lies on SS. By fixing . By fixing yy = = bb, we are , we are restricting our attention to the curve restricting our attention to the curve CC11
in which the vertical plane in which the vertical plane yy = = bb intersects intersects SS..
Interpretations (cont’d)Interpretations (cont’d)
Likewise, the vertical plane Likewise, the vertical plane xx = = aa intersects intersects SS in a curve in a curve CC22..
Both of the curves Both of the curves CC11 and and CC22 pass pass
through the point through the point PP.. This is illustrated on the next slide:This is illustrated on the next slide:
Interpretations (cont’d)Interpretations (cont’d)
Interpretations (cont’d)Interpretations (cont’d)
Notice that…Notice that… CC11 is the graph of the function is the graph of the function gg((xx) = ) = ff((xx, , bb), so ), so
the slope of its tangent the slope of its tangent TT11 at at PP is is gg′(′(aa) = ) = ffxx((aa, ,
bb);); CC22 is the graph of the function is the graph of the function GG((yy) = ) = ff((aa, , yy), so ), so
the slope of its tangent the slope of its tangent TT22 at at PP is is GG′(′(bb) = ) = ffyy((aa, ,
bb).).
Thus Thus ffxx and and ffyy are the are the slopes of the tangent slopes of the tangent
lineslines at at PP((aa, , bb, , cc) to ) to CC11 and and CC22..
Interpretations (cont’d)Interpretations (cont’d)
Partial derivatives can also be Partial derivatives can also be interpreted as interpreted as rates of changerates of change..
If If zz = = ff((xx, , yy), then…), then… ∂∂zz/∂/∂xx represents the rate of change of represents the rate of change of zz
with respect to with respect to xx when when yy is fixed. is fixed. Similarly, ∂Similarly, ∂zz/∂/∂yy represents the rate of represents the rate of
change of change of zz with respect to with respect to yy when when xx is is fixed.fixed.
Interpret these numbers as slopes.Interpret these numbers as slopes. SolutionSolution We have We have
Solution (cont’d)Solution (cont’d)
The graph of The graph of ff is the paraboloid is the paraboloidzz = 4 – = 4 – xx22 – 2 – 2yy22 and the vertical plane and the vertical plane yy = 1 = 1 intersects it in the parabola intersects it in the parabola zz = 2 – = 2 – xx22, , yy = 1. = 1.
The slope of the tangent line to this parabola The slope of the tangent line to this parabola at the point (1, 1, 1) is at the point (1, 1, 1) is ffxx(1, 1) = –2.(1, 1) = –2.
Similarly, the plane Similarly, the plane xx = 1 intersects the = 1 intersects the graph of graph of ff in the parabola in the parabola zz = 3 – 2 = 3 – 2yy22, , xx = 1. = 1.
Solution (cont’d)Solution (cont’d)
The slope of the tangent line to this The slope of the tangent line to this parabola at the point (1, 1, 1) is parabola at the point (1, 1, 1) is ffyy(1, (1,
1) = –4:1) = –4:
ExampleExample
IfIf
SolutionSolution Using the Chain Rule for Using the Chain Rule for functions of one variable, we havefunctions of one variable, we have
. and calculate ,1
sin,yf
xf
yx
yxf
ExampleExample
Find ∂Find ∂zz/∂/∂xx and ∂ and ∂zz/∂/∂yy if if zz is defined is defined implicitly as a function of implicitly as a function of xx and and yy by by
xx33 + + yy33 + + zz33 + 6 + 6xyzxyz = 1 = 1 SolutionSolution To find ∂ To find ∂zz/∂/∂xx, we , we
differentiate implicitly with respect differentiate implicitly with respect to to xx, being careful to treat , being careful to treat yy as a as a constant:constant:
Solution (cont’d)Solution (cont’d)
Solving this equation for ∂Solving this equation for ∂zz/∂/∂xx, we , we obtainobtain
Similarly, implicit differentiation Similarly, implicit differentiation with respect to with respect to yy gives gives
More Than Two VariablesMore Than Two Variables Partial derivatives can also be defined Partial derivatives can also be defined
for functions of three or more for functions of three or more variables, for examplevariables, for example
If If ww = = ff((xx, , yy, , zz), then ), then ffxx = ∂ = ∂ww/∂/∂xx is the is the
rate of change of rate of change of ww with respect to with respect to xx when when yy and and zz are held fixed. are held fixed.
More Than Two Variables More Than Two Variables (cont’d)(cont’d)
But we can’t interpret But we can’t interpret ffxx
geometrically because the graph of geometrically because the graph of ff lies in four-dimensional space.lies in four-dimensional space.
In general, if In general, if uu = = ff((xx11, , xx2 2 ,…, ,…, xxnn), then), then
and we also writeand we also write
ExampleExample
Find Find ffxx ,, f fy y , and , and ffzz if if ff((xx, , yy, , zz) = ) = eexyxy ln ln zz..
SolutionSolution Holding Holding yy and and zz constant and constant and differentiating with respect to differentiating with respect to xx, we , we havehave
ffxx = = yeyexyxy ln ln zz
SimilarlySimilarly,,
ffyy = = xexexyxy ln ln zz and and ffzz = = eexyxy//zz
Higher DerivativesHigher Derivatives
If If ff is a function of two variables, then is a function of two variables, then its partial derivatives its partial derivatives ffxx andand f fyy are also are also
functions of two variables, so we can functions of two variables, so we can consider their partial derivativesconsider their partial derivatives
((ffxx))xx ,, ((ffxx))yy ,, ((ffyy))x x , and (, and (ffyy))y y ,,
which are called the which are called the second partial second partial derivativesderivatives of of ff..
Partial derivatives occur in Partial derivatives occur in partial partial differential equationsdifferential equations that express that express certain physical laws.certain physical laws.
For instance, the partial differential For instance, the partial differential equationequation
is called is called Laplace’s equationLaplace’s equation..
Solutions of this equation are called Solutions of this equation are called harmonic functionsharmonic functions and play a role and play a role in problems of heat conduction, fluid in problems of heat conduction, fluid flow, and electric potential.flow, and electric potential.
For example, we can show that the For example, we can show that the function function uu((xx, , yy) = ) = eexx sin sin yy is a is a solution of Laplace’s equation:solution of Laplace’s equation:
describes the motion of a waveform, describes the motion of a waveform, which could be an ocean wave, which could be an ocean wave, sound wave, light wave, or wave sound wave, light wave, or wave traveling along a string.traveling along a string.
For instance, if For instance, if uu((xx, , tt) represents the ) represents the disaplacement of a violin string at time disaplacement of a violin string at time tt and at a distance and at a distance xx from one end of from one end of the string, then the string, then uu((xx, , tt) satisfies the ) satisfies the wave equation. (See the next slide.)wave equation. (See the next slide.)
Here the constant Here the constant aa depends on the depends on the density of the string and on the tension density of the string and on the tension in the string.in the string.
So So uu satisfies the wave equation. satisfies the wave equation.
ReviewReview
Definition of partial derivativeDefinition of partial derivative Notations for partial derivativesNotations for partial derivatives Finding partial derivativesFinding partial derivatives Interpretations of partial derivativesInterpretations of partial derivatives Function of more than two variablesFunction of more than two variables Higher derivativesHigher derivatives Partial differential equationsPartial differential equations