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Section 12.1 Partial derivatives
Objectives
Measure how a multivariable function changes when only one input varies.
Interpret partial derivatives visually using traces with one variable held constant.
Compute partial derivatives by treating other variables as fixed and applying single-variable rules.
Start with windchill example? Or maybe business example?
If
\(z=f(x,y)\) is a function, then the partial derivatives are defined as follows:
\begin{align*}
\pdv{z}{x} \amp = f_{x}(x,y) = \limit{h\to 0}\frac{f(x+h,y)-f(x,y)}{h} \\
\pdv{z}{y} \amp = f_{y}(x,y) = \limit{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}
\end{align*}
This represents the
sensitivity of
\(z\) to changes in
\(x\) or
\(y\) alone .
Geometrically, this is like taking a trace with a constant
\(x\) or
\(y\text{.}\)
To compute the partial derivative with respect to one variable, treat the other variable as constant and take the single-variable derivative.
Might it be a good idea to actually numerically set a variable equal to a few different constants first, and only then treat the variable as an arbitrary constant?
Example:
\(f(x,y)=x^{2}-3xy+y^{3}\)
\(f(x,1)=x^{2}-3x+1\text{,}\) so
\(\left.\pdv{f}{x}\right|_{y=1}=2x-3\text{.}\)
\(f(x,2)=x^{2}-6x+8\text{,}\) so
\(\left.\pdv{f}{x}\right|_{y=2}=2x-6\text{.}\)
\(f(x,3)=x^{2}-9x+27\text{,}\) so
\(\left.\pdv{f}{x}\right|_{y=3}=2x-9\text{.}\)
In general,
\(\pdv{f}{x}=2x-3y\text{.}\)
\(f(1,y)=1-3y+y^{3}\text{,}\) so
\(\left.\pdv{f}{y}\right|_{x=1}=-3+3y^{2}\text{.}\)
\(f(2,y)=4-6y+y^{3}\text{,}\) so
\(\left.\pdv{f}{y}\right|_{x=2}=-6+3y^{2}\text{.}\)
\(f(3,y)=9-9y+y^{3}\text{,}\) so
\(\left.\pdv{f}{y}\right|_{x=3}=-9+3y^{2}\text{.}\)
In general,
\(\pdv{f}{y}=-3x+3y^{2}\text{.}\)
Three-variable example of some kind
