# What do the symbols d/dx and dy/dx mean?

Okay this may sound stupid but I need a little help... What do $\Large \frac{d}{dx}$ and $\Large \frac{dy}{dx}$ mean?

I need a thorough explanation. Thanks.

• This is not precalculus, it is calculus. These symbols are derivatives. Are you familiar with derivatives?\ Commented Mar 25, 2013 at 15:44

The symbol $$\frac{dy}{dx}$$ means the derivative of $y$ with respect to $x$. If $y = f(x)$ is a function of $x$, then the symbol is defined as $$\frac{dy}{dx} = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}.$$ and this is is (again) called the derivative of $y$ or the derivative of $f$. Note that it again is a function of $x$ in this case. Note that we do not here define this as $dy$ divided by $dx$. On their own $dy$ and $dx$ don't have any meaning (here). We take $\frac{dy}{dx}$ as a symbol on its own that can't be slit up into parts.

The symbol $$\frac{d}{dx}$$ you can consider as an operator. You can apply this operator to a (differentiable) function. And you get a new function. So if $f$ is a (differentiable) function that it makes sense to "apply" $\frac{d}{dx}$ to $f$ and write $$\frac{d}{dx}f$$ If you write $y = f(x)$, then this is the same as $$\frac{d}{dx}y = \frac{dy}{dx}.$$

• The confusion often arises from the fact that many writers call $dy/dx$ a "symbol" as if it were atomic, but then later start doing algebra with it. This leads to the question, "well, then what is $dy$ really?" Commented Mar 25, 2013 at 15:55
• @Fixee: That is right. I often see this confusion which is why I always emphasize that $\frac{dy}{dx}$ is just a symbol. It is not a fraction. Commented Mar 25, 2013 at 15:57
• @Thomas It's kinda a fraction. It's the limit of a fraction. Commented May 1, 2015 at 21:35
• (Or the "standard part" of a fraction, if you're doing nonstandard analysis...) Commented May 1, 2015 at 21:35
• @Fixee See my answer. It is nevertheless CAN be defined as a fraction of two functions, rather than an atomic object. Commented May 1, 2015 at 21:59

$$\frac{d}{dx}$$ means differentiate with respect to $$x$$.

$$\frac{dy}{dx}$$ means differentiate $$y$$ with respect to $$x$$.

Do you have any concrete examples for which you need to calculate these two? It would probably make it more easy to grasp for you if I could explain it in a few examples.

• Maybe calculating the flow rate of a tap based on how the container gets filled? Commented Mar 23, 2023 at 7:57

$d f$ means the differential of function $f$. By definition $(df)(x) = \lambda t\in\mathbb{R}:f'(x)\cdot t$. In other words, differential is the linear function (of an additional variable denoted $t$ here) whose tangent is the derivative of $f$.

$d$ alone means the differential operator (a function of argument $f$).

Exercise: Show that $\frac{df}{dx}=f'$.

• Note that $\frac{F}{G}$ is defined for two functions $F$ and $G$ as $\frac{F}{G}(x)=\frac{F(x)}{G(x)}$. Thus $\frac{df}{dx}$ makes sense. Commented May 1, 2015 at 21:16
• It is described in en.wikipedia.org/wiki/Differential_of_a_function Commented May 2, 2015 at 19:33

I like to look at it this way: $$dx$$ and $$dy$$ are just representations of change in accordance to either $$x$$ or $$y$$ axis. If you take the the symbol for derivative $$\frac{dy}{dx}$$ and compare it to the formula for the slope: $$\frac{f(x_1) - f(x_2)}{x_1 - x_2}$$ we can clearly see that $$dy$$ and $$dx$$ depict change in $$y$$ and change in $$x$$ respectively.

If $$y=f(x)$$ i.e., where $$y$$ is the equation ( the dependent variable) and $$x$$ is the independent variable. Meaning $$x$$ changes $$y$$.

Now $$\frac{dy}{dx}$$ means differentiate the equation $$y$$ in respect to $$x$$.

$$\frac{d}{dx}$$ means differentiate in respect to $$x$$.

Same way $$\log x$$ means find the natural logarithm of $$x$$, $$\frac{d}{dx} x$$ means find the derivative of $$x$$.

N.B. In an equation $$k= h²+5$$, $$\frac{dk}{dh}$$ means differentiate the equation $$k$$ in respect to $$h$$. Its not always $$\frac{dy}{dx}$$.

I hope you understand

• Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. Commented Jun 27, 2019 at 3:57
• See the answer given by Thomas Commented Jun 27, 2019 at 4:52