# Derivatives and rates of change

## Slopes of lines and tangent lines

- Recall slopes
- Let \(m\) be a number, and \(P(a,c)\) a plane point.
- Then the equation of the line through \(P\) with slope \(m\) is the following. \[
y-c=m(x-a)
\]
- How you can remember this is that \(y=mx\) is the line with slope \(m\) through the origin \(O(0,0)\), which you then need to translate by \(c\) upwards, and by \(a\) to the right.

- Tangent lines
- Let \(f(x)\) be a function, and let \(C:y=f(x)\) be the curve which is its graph.
- Let us fix two numbers \(a\) and \(b\) such that the closed interval \([a,b]\) is contained in the domain of \(f(x)\), and consider the points \(P(a,f(a))\) and \(Q(b,f(b))\) on \(C\).
- Then the line between \(P\) and \(Q\) will have the following slope. \[
m_{PQ}=\frac{f(b)-f(a)}{b-a}.
\]
*Definition*. Let \(f(x)\) be a function, and \(a\) a point in its domain. Then the *tangent line* to the curve \(C=f(x)\) at \(P(a,f(a))\) is the line through \(P\) with slope \[
m=\lim_{b\to a}\frac{f(b)-f(a)}{b-a},
\] provided that the limit exists.
- Click here for an illustration.
- Exercise. 2.7: 4aib. Click here for part c

## Rate of change and derivatives

- Let \(f(x)\) be a function, and \(a\) and \(b\) two numbers such that the closed interval \([a,b]\) is contained in the domain of \(f(x)\).
- We can interpret the difference \(f(b)-f(a)\) as the
*average rate of change* of \(f(x)\) on the interval \([a,b]\).
- There's another notation emphasising this point of view.
- The change in \(x\) is \(\Delta x=b-a\), and the change in \(y=f(x)\) is \(\Delta y=f(b)-f(a)\).
- Then the average rate of change can be written as \(\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}\).

- As we make \(b\to a\), we get the
*(instantaneous) rate of change at \(x=a\)*.
- Since we make \(b\to a\), it's better to write \(b=a+h\).
- Note that \(h=\Delta x=b-a\).
*Definition.* Let \(f(x)\) be a function, and \(a\) a number such that the domain of \(f(x)\) contains some open interval around \(a\). Then the *derivative of \(f(x)\) at \(x=a\)* is \[
f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h},
\] provided that the limit exists.
- An alternative notation is \(\frac{\mathrm df}{\mathrm dx}\big|_{x=a}\).
- Exercises. 2.7: 8ab, 10ab, 14, 20, 22, 60