Partial differentiation

Partial derivatives help us analyse the influence of each individual variable on a system. This is crucial for understanding surfaces and optimisation problems in fields like engineering, physics and economics. Use this resource to learn how to find partial derivatives.

Partial derivatives reveal how a function with many variables changes when you adjust just one of the variables in the input. Consider the following equation:
\[z=x^{2}+4y^{2}\]

Here, we have two independent variables, so the function is \(f(x,y)\). The graph exists in a three-dimensional space.

When we differentiate such a function, we need to be able to differentiate \(z\) with respect to either \(x\) or \(z\). This involves treating the other variable as a constant. We calculate the partial derivatives:

• \(\dfrac{\partial z}{\partial x}\), the partial derivative of \(z\) with respect to \(x\) where we treat \(y\) as a constant
• \(\dfrac{\partial z}{\partial y}\), the partial derivative of \(z\) with respect to \(y\) where we treat \(x\) as a constant.

The symbol we use to denote a partial differential is the curve d, \(\partial\), often called "del".

An alternative notation for partial derivatives is \(f_{x}\) for \(\dfrac{\partial f}{\partial x}\)and \(f_{y}\) for \(\dfrac{\partial f}{\partial y}\).

Example 1 – evaluating partial derivatives

If \(z=x^{2}+4y^{2}\), find \(\dfrac{\partial z}{\partial x}\) and \(\dfrac{\partial z}{\partial y}\).

\(\dfrac{\partial z}{\partial x}\) is the partial derivative of \(z\) with respect to \(x\), treating \(y\) as a constant.
\[\begin{align*} \frac{\partial z}{\partial x} & = 2x+0\\
& = 2x
\end{align*}\]

\(\dfrac{\partial z}{\partial y}\) is the partial derivative of \(z\) with respect to \(y\), treating \(x\) as a constant.
\[\begin{align*} \frac{\partial z}{\partial y} & = 0+8y\\
& = 8y
\end{align*}\]

Evaluating partial derivatives at a point

We can find partial derivatives at particular points. For example, \(f_{x}(2,1)\) refers to the value of the partial derivative of \(f\) with respect to \(x\) at the point where \(x=2\) and \(y=1\).

Example – evaluating partial derivatives at a point

If \(f(x,y)=2x^{3}y+3y^{2}\), find \(f_{y}(1,3)\).

\(f_{y}\) indicates that we are looking for the derivative of \(f\) with respect to \(y\), treating \(x\) as a constant.
\[f_{y} = 2x^{3}+6y\]

We can then substitute \(x=1\) and \(y=3\) into \(f_{y}\) to find the partial derivative at that point.
\[\begin{align*} f_{y}(1,3) & = 2(1)^{3}+6(3)\\
& = 2+18\\
& = 20
\end{align*}\]

Higher order partial derivatives

A function \(z=f(x,y)\) has two partial derivatives: \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\). This kind of function will have four second order partial derivatives.

1. Differentiating with respect to \(x\), then with respect to \(x\) gives us:
   \[\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial^{2}f}{\partial x^{2}}=f_{xx}\]
2. Differentiating with respect to \(y\), then with respect to \(y\) gives us:
   \[\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right)=\frac{\partial^{2}f}{\partial y^{2}}=f_{yy}\]
3. Differentiating with respect to \(x\), then with respect to \(y\) gives us:
   \[\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial^{2}f}{\partial x\partial y}=f_{xy}\]
4. Differentiating with respect to \(y\), then with respect to \(x\) gives us:
   \[\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)=\frac{\partial^{2}f}{\partial y\partial x}=f_{yx}\]

\(f_{xy}\) is always equal to \(f_{yx}\).

Example – evaluating higher order partial derivatives

Find all second order partial derivatives of \(f(x,y)=x^{2}y^{3}+2x\cos(y)\).

First, we need to find the first order partial derivative with respect to \(x\) and \(y\).
\[f_{x}=2xy^{3}+2\cos(y)\]\[f_{y}=3x^{2}y-2x\cos(y)\]

The second order partial derivatives of \(f(x,y)\) are \(f_{xx}\), \(f_{yy}\) and \(f_{xy}=f_{yx}\).
\[f_{xx}=2y^{3}\] \[f_{yy}=6x^{2}y-2x\sin(y)\] \[f_{xy}=f_{yx}=6xy^{2}-2\sin(y)\]

Exercise – finding partial derivatives

1. Find the partial derivatives \(\dfrac{\partial z }{\partial x}\) and \(\dfrac{\partial z }{\partial y}\) for each of the following.
1. \(z=3x^{4}+2y^{3}\)
2. \(z=x^{2}y\)
3. \(z=3xe^{2y}\)
4. \(z=\ln(x^{3}y^{5}-2)\)
2. Find the value of the indicated partial derivative at the given point.
1. \(f_{x}(2,3)\) where \(f(x,y)=x^{4}-4y^{2}\)
2. \(f_{y}(-1,1)\) where \(f(x,y)=\ln(x^{2}+y^{3})\)
3. Find the first and second order partial derivatives of the following.
1. \(f(x,y)=x\ln(y)\)
2. \(f(x,y)=x^{3}+x^{2}y-3xy^{2}+y^{3}\)
3. \(f(x,y)=\sin(xy)\)
4. \(f(x,y)=x\cos(y)+ye^{x}\)

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