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.
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$\begingroup$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 $$ \frac{d}{dx}y = \frac{dy}{dx}. $$
$\endgroup$ 5 $\begingroup$$\frac{d}{dx}$ means differentiate with respect to $x$.
$\frac{dy}{dx}$ means differentiate $y$ with respect to $x$.
Do you have any concrete examples for which you need to calculate these two? It would probably make it more easy to grasp for you if I could explain it in a few examples.
$\endgroup$ $\begingroup$$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 $\frac{df}{dx}=f'$.
$\endgroup$ 2 $\begingroup$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
$\endgroup$ 2 $\begingroup$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.
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