## What Does This Wobbly “d” Do?

What is the difference between $$\frac{d}{dx}$$ and $$\frac{\partial}{\partial x}$$?

…is a question I get surprisingly often when tutoring friends.

Short version: The difference is all about dependency!

The “regular d” in $$\frac{d}{d x}$$ denotes ordinary differentiation: assumes all variables are dependent on $$x$$ ($$\rightarrow$$ envoke chain/product rule to treat the other variables as functions of $$x$$).

The “wobbly d” in $$\frac{\partial}{\partial x}$$ denotes partial differentiation and assumes that all variables are independent of $$x$$.

You can make a straight d from a wobbly d by using a beautiful thing:

$$\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\frac{dy}{dx} + \frac{\partial f}{\partial z}\frac{dz}{dx}$$

You can remember it this way: the partial derivative is partially a “d”, or the the “wobbly d” is partial to nemself and bends nir neck down to look at nirs reflection $$\partial$$.

Sidenote: Ne/nem/nir/nirs/nemself is a pretty swell gender neutral pronoun set!

Longer version: Before we start – what is a derivative, anyway?

The derivative of a function at a chosen point describes the linear approximation of the function near that input value. Recall the trusty formula: $$\Delta y = m\Delta x$$, where m is the slope and $$\Delta$$ represents the change in the variable? For a (real-valued) function of a (real) single variable, the derivative at that point is = tangent line to the graph at that point.

The beauty of math is: $$m = \frac{\Delta y}{\Delta x} \equiv$$ “How much one quantity (the function) is changing in response to changes in another quantity ($$x$$) at that point (it’s input, assuming $$y(x)$$).”

So, it makes sense that derivative of any constant is 0, since a constant (by definition) is constant $$\rightarrow$$ unchanging!

$$\frac{d(c)}{dx}=0$$, where $$c$$ is any constant.

What about everything that isn’t a constant?

That means, $$\frac{d(x^2)}{dx} = 2x$$, since $$x^2$$ is dependent on $$x$$. $$\frac{d}{dx}$$ denotes ordinary differentiation, i.e. all variables are dependent on the given variable (in this case, $$x$$).

But what about $$\frac{d(y)}{dx}$$? Looking at this equation, we immediately assume $$y$$ is a function of $$x$$. Otherwise, it makes no sense. $$\frac{d(y)}{dx} \equiv \frac{d(y(x))}{dx}$$

On the other hand, $$\frac{\partial (y)}{\partial x}$$ denotes partial differentiation. In this case, all variables are assumed to be independent.

$$\frac{\partial (y)}{\partial x} = 0$$

Let’s compare them with an example. $$f(x,y) = ln(x)sec(y) + y$$

$$\frac{\partial f}{\partial x} = \frac{sec(y)}{x}$$

$$\frac{df}{dx}$$ implies that $$y$$ is dependent(a function of) $$x$$, i.e. $$f = (ln(x)sec(y(x)) + y(x))$$

$$\frac{df}{dx} = \frac{dy}{dx}(ln(x)tan(y(x))sec(y(x)) + 1) + \frac{sec(y(x))}{x}$$

As you can see, $$\frac{\partial f}{\partial x} \neq \frac{df}{dx}$$.

By the way, to use LaTex in Blogger include the following before </head> in your Template (source):

<script type="text/x-mathjax-config"> MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['$$','$$']]}}); </script>
<script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"> </script>