A Guide to Partial Derivatives: Topics of Partial Derivatives

Setting the Ground on Partial Derivatives

Our series which elaborates on the topic of partial derivatives began first by introducing functions of multiple variables. An understanding of how these functions operate is integral to executing partial derivatives. If you need a refresher on this subject before embarking further, check out this article. Furthermore, limits are an essential topic to review prior to partial derivatives, and you can find our take on this subject here. With that, let us begin this guide to taking partial derivatives.

Relating Multi-Variable Functions to Partial Derivatives

Think back to our functions of multiply variables which take the form f(x,y)=z. Sometimes, we may want to observe only how only one particular variable affects the function’s behavior alone. In this case, we may want to let one of these variables vary while one variable remains constant. For example, if we would like to observe function behavior of the function as x varies and y remains constant, we first say y=b where ‘b’ is a constant. In this case, we are really considering a new function of the form g(x)=f(x,b). If the function ‘g’ has a derivative at the point ‘a’, then we might say that the derivative of ‘g’ is a partial derivative of ‘f’ at (a,b). We can model this rule as:

f_x(a,b)=g'(a)\enspace \text{where}\enspace g(x)=f(x,b)

Proving Partial Derivatives with Limits

Here is where the importance of understanding limits becomes apparent. We can use our understanding of limits to procure a much more robust conceptualization of partial derivatives. Recall that our definition of limits as they pertain to derivatives takes the form:

g'(a)=\lim\limits_{h \to 0}\frac{g(a+h)-g(a)}{h}

However, this definition only works for derivatives of functions with a single variable. Therefore, we can modify this to work for a function consisting of two variables, particularly when one of the variables is set to the constant ‘b’. The conversion is rather simple, and takes the form:

f_x(a,b)=\lim\limits_{h \to 0}\frac{f(a+h,b)-f(a,b)}{h}

We can similarly acquire the derivative of ‘f’ with respect to ‘y’ at (a,b) through a similar form:

f_y(a,b)=\lim\limits_{h \to 0}\frac{f(a,b+h)-f(a,b)}{h}

We may sum up the rules for partial derivatives as follows. If a function ‘f’ is comprised of two variables ‘x’ and ‘y’, then the partial derivatives are defined by the functions fx and fy such that:

f_x(x,y)=\lim\limits_{h \to 0}\frac{f(x+h,y)-f(x-y)}{h} \newline f_y(x,y)=\lim\limits_{h \to 0}\frac{f(x,y+h)-f(x-y)}{h} \newline

Notations and Rules for Partial Derivatives

There are several different notations to indicate the taking of a partial derivative, and they are really up to the user which one you prefer. If we have a function of the form f(x,y)=z, then we can represent the taking of a partial derivative with respect to ‘x’ or ‘y’ as:

f_x(x,y)=f_x=\frac{\partial f}{\partial x}=\frac{\partial}{\partial x}f(x,y)=\frac{\partial z}{\partial x}\newline \cdot \newline f_y(x,y)=f_y=\frac{\partial f}{\partial y}=\frac{\partial}{\partial y}f(x,y)=\frac{\partial z}{\partial y}

We can also simplify the taking of partial derivatives for a function down to two simple rules:

  1. If we are taking the partial derivative of f(x,y) such that we are looking for fx, then keep y constant and differentiate f(x,y) with respect to x.
  2. If we are taking the partial derivative of f(x,y) such that we are looking for fy, then keep x constant and differentiate f(x,y) with respect to y.

With these two rules in mind, taking the partial derivatives of a multi-variable function is made significantly easier.

Example of Partial Derivatives

Now that the methodology of taking partial derivatives has been made clear, let us attempt a practical approach of this technique. Let us do so by considering a novel function and taking its partial derivatives. Have a look at the following function:

f_x(x,y)=x^3+x^2y^3-2y^2

With this, let us attempt to find the partial derivatives of the function with respect to ‘x’ and ‘y’ at the point (2,1). Let’s begin holding ‘y’ constant and differentiating with respect to ‘x’. Doing so we find:

f_x(x,y)=3x^2+2xy^3\newline f_x(2,1)=3(2)^2+2(2)(1^3)\newline f_x(2,1)=16

This result demonstrates that the derivative of the function with respect to ‘x’ yields 16. Now, what does this mean? Well, recall that our function ‘fx‘ is actually a novel function separate from the unadulterated one. This, like all other derivates, yields a function that reflects a rate of change for the function at the point (2,1), but only the rate of change with respect to ‘x’. If we want to consider ‘y’, let us hold ‘x’ constant and differentiate with respect to ‘y’. Doing so produces:

f_y=3x^2y^2-4y\newline f_y(2,1)=(3)(2^2)(1^2)-4(1)\newline f_y(2,1)=8

So, what can we say about this? Again, a derivative, as always, reflects rate of change. From this, we can comfortably state that the rate of change with respect to ‘x’ is twice as much the rate of change with respect to ‘y’. We expand more upon the interpretations of derivatives in the subsequent section.

Geometric Understanding of Partial Derivatives

Remember that a function of the form z=f(x,y) represents a surface, which we can call ‘S’. If the function f(a,b)=c, then the point (a,b,c) must lie on this surface S. If we take the partial derivative of ‘f’ with respect to ‘x’, then we must hold ‘y’ constant such that we execute fx(x,b). Now, because ‘y’ has been restricted, the derivative reflects the rate of change at the point (a,b,c) in the x/z-dimension. Recall from the beginning that taking a partial derivative of a function is really like creating a new equation, so we could say that the function g(x)=f(x,b). If this is true, then the function g(x) returns a line along the surface ‘S’ which intersects the point (a,b,c). Now, taking the derivative of this function finds the rate of change in the x/z-dimension. So, if we have g'(x)=fx(x,b), then g'(x) represents the tangent of the surface at point ‘c’. The same can be said in the case of ‘y’. Consider this geometric representation in the image below:

The Take Away

Hopefully this guide has served well to demonstrate the utility of partial derivatives, and although the form of these functions is rather different than what you may be used to, it is actually quite simple. Up to this point, we have now examined not only taking partial derivatives, but the functions of multiple variables which constitute them as well as their limits and continuity. If you have not referenced these I will render them below. Consider other resources on partial derivatives as well, including the following. As this series continues, we will build upon the knowledge we have established here. We will explore several nuances of partial derivatives, including the chain rule and tangent planes.

Modeling Functions of Multiple Variables

Limits and Continuity of Higher Variable Functions

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