An introduction to partial derivatives and the heat equation
Daniel Bilalov
Defining a partial derivative
Partial derivatives are derivatives of a function made up of two or more variable with respect to one of those variables while keeping the rest constant. All smooth (infinitely differentiable) functions have a Taylor series expansion 1 , the expansion of 𝑓(𝑥 + 𝑑𝑥, 𝑦) is given below:
𝜕 2 𝑓 𝜕𝑥 2
𝜕 3 𝑓 𝜕𝑥 3
𝜕𝑓 𝜕𝑥
(𝑥, 𝑦)𝑑𝑥 2 +
(𝑥, 𝑦)𝑑𝑥 3 …
(𝑥, 𝑦)𝑑𝑥 +
𝑓(𝑥 + 𝑑𝑥, 𝑦) = 𝑓(𝑥, 𝑦) +
After some algebraic manipulation we are left with the equation below:
𝜕 2 𝑓 𝜕𝑥 2
𝜕 3 𝑓 𝜕𝑥 3
𝑓(𝑥 + 𝑑𝑥, 𝑦) − 𝑓(𝑥, 𝑦) 𝑑𝑥
𝜕𝑓 𝜕𝑥
(𝑥, 𝑦)𝑑𝑥 2 …
(𝑥, 𝑦)+
(𝑥, 𝑦)𝑑𝑥 +
=
Taking the limit 𝑑𝑥 →0 on both sides gives the following:
𝜕 2 𝑓 𝜕𝑥 2
𝜕 3 𝑓 𝜕𝑥 3
𝑓(𝑥 + 𝑑𝑥, 𝑦) − 𝑓(𝑥, 𝑦) 𝑑𝑥
𝜕𝑓 𝜕𝑥
(𝑥, 𝑦)𝑑𝑥 2 …)
(𝑥, 𝑦)+
(𝑥, 𝑦)𝑑𝑥 +
lim 𝑑𝑥→0
= lim 𝑑𝑥→0
(
𝑓(𝑥 + 𝑑𝑥, 𝑦) − 𝑓(𝑥, 𝑦) 𝑑𝑥
𝜕𝑓 𝜕𝑥
(𝑥, 𝑦) = 𝑓 𝑥
lim 𝑑𝑥→0
=
This is the formal definition of a partial derivative and we also take the opportunity to introduce another notation for it. Below is demonstrated how to find the partial derivatives for a function 𝑓(𝑥, 𝑦)
𝑓(𝑥, 𝑦) = 𝑥 2 ln(𝑦)+𝑒 𝑥 sin(𝑦)
𝑥 2 𝑦
𝜕𝑓 𝜕𝑥
𝜕𝑓 𝜕𝑦
= 2𝑥 ln(𝑦) + 𝑒 𝑥 sin(𝑦) ,
+𝑒 𝑥 cos(𝑦)
𝑓 𝑥 =
𝑓
𝑦 =
=
Interpretations of partial derivatives
Previously via differentiation we could find equations of lines tangent to curves at certain points, now with the use of partial derivatives, we can find the same for curves (and shapes) in 3 dimensions. Take the graph with equation 𝑧(𝑥, 𝑦) = 10 − 4𝑥 2 −𝑦 2 where 𝑥, 𝑦 are independent variables. Below is demonstrated how to find the equations of the lines tangent to 𝑧(𝑥, 𝑦) at the point (1,2) for the plane 𝑥 =1 and 𝑦 =2 .
𝜕𝑧 𝜕𝑥
𝜕𝑧 𝜕𝑦
=−8𝑥,
=−2𝑦
1 Bronson and Costa (2014), 262.
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