Leibniz Notation

The Leibniz notation is where we denote the derivative of the function $f(x)$ by
\[
\frac{df}{dx}.
\]
In this lesson we explain where this notation comes from.

The use of the greek "$\Delta$" is common in science and means "take the difference" or equivalently "the change in". So if I had two variables, $a_1$ and $a_2$, and took the difference $a_2-a_1$, then this difference is typically denoted by $\Delta a$. In other words, $\Delta a=a_2-a_1$. Or in plan English, "the change in $a$ is $a_2-a_1$."

Recall that a function's derivative is defined as the limit
\[
f'(x)=\lim_{h\to 0}\,\frac{f(x+h)-f(x)}{h}.
\]

Observe that since $h$ equals $x+h-x$, it is the difference between two values on the x-axis: $x+h$ and $x$. Using our "$\Delta$" notation we have $\Delta x=(x+h)-x=h$. Furthermore, we denote $f(x+h)-f(x)$ by $\Delta f$. With this in mind we can now denote the derivative as
\[
f'(x)=\lim_{\Delta x\to 0}\,\frac{\Delta f}{\Delta x}.
\]

Now here is the thing to remember. When the difference becomes extremely small, we stop denoting the difference by "$\Delta$" and instead denote it by "$d$". In plain English, $\Delta$ is a difference and $d$ is a very small difference. So $\Delta a=a_2-a_1$, but when $a_2-a_1$ becomes very small we switch to $da$. We can use our limit notation to say that
\[
da=\lim_{\Delta a\to 0}\,\Delta a.
\]
Similarly, we can say that

This is where the Leibniz notation comes from.

Finally, the Leibniz notation allows us to remember a very important concept. Remember that for a straight line $f(x)=mx+b$. By knowing the slope of the straight line and its value at some point $x_1$, we can find how much the function changes when we move from $x_1$ to $x_2$. Let's do some quick and easy algebra to illustrate:
\begin{eqnarray}
f(x_2)-f(x_1)&=&(mx_2+b)-(mx_1+b)\\
f(x_2)-f(x_1)&=&m(x_2-x_1)\\
\Delta f&=&m\Delta x.
\end{eqnarray}
By using our notation of $df/dx$ for the slope we can rewrite it as
\[
\Delta f=\frac{df}{dx}\cdot\Delta x.
\]

Now for a function that is not a straight line we have

Notice that we replaced the "$=$" sign by the approximation symbol "$\approx$". The graph below illustrates the approximation to $\Delta f$ by $(df/dx)\Delta x$.

So here is where the Leibniz notation helps us remember this very important approximation. If we think of $\Delta f\approx df$ and $\Delta x\approx dx$ then
\[
1\approx\frac{\Delta x}{dx}
\]
and since $\Delta f\approx df$ we have
\begin{eqnarray}
\Delta f&\approx& df\\
\Delta f&\approx& df\cdot 1\\
\Delta f&\approx& df\cdot\frac{\Delta x}{dx}\\
\Delta f&\approx& \frac{df}{dx}\cdot\Delta x.\\
\end{eqnarray}

So the Leibniz Notation helps us remember two important things:
\[
\frac{df}{dx}=\lim_{h\to 0}\,\frac{f(x+h)-f(x)}{h}\quad\mbox{and}\quad\Delta f\approx\frac{df}{dx}\cdot\Delta x.
\]
Just be careful not to think of this as actual math. The actual mathematics is in the function limits. The Leibniz notation just gives a simple algebraic way of remembering these two important concepts.