The average rate-of-change of a continuous function \(f\)
between \(x=a\) and \(x=b\) is \[ \frac{f(b)-f(a)}{b-a}\,. \]
We also call it the average rate-of-change on the interval \([a,b].\)
On the graph of \(f\) this is the slope of the line that passes through
the points \(\bigl(a, f(a)\bigr)\) and \(\bigl(b, f(b)\bigr),\)
referred to as the secant line through these points.
The slope of this secant line and the average rate-of-change of \(f\)
are only equal if \(f\) is continuous between \(a\) and \(b.\)
Same idea from a different perspective:
the average rate-of-change of a function \(f\)
on an interval of length \(h\) after some initial point \(x=c\) is
\[ \frac{f(c+h)-f(c)}{h}\,. \]
This expression is called the difference quotient of \(f\) at \(c.\)
On the graph of \(f\) this is the slope of the secant line that passes through
the points \(\bigl(c, f(c)\bigr)\) and \(\bigl(c+h, f(c+h)\bigr).\)
The instantaneous rate-of-change of a continuous function \(f\)
at \(x=c\) is \[ \lim\limits_{h\to 0}\frac{f(c+h)-f(c)}{h}\,. \]
On the graph of \(f\) this is the slope of the line
that just barely touches point \(\bigl(c, f(c)\bigr)\)
referred to as the tangent line at this point.
Similarly we say that the line is tangent to the graph at \(x=c\)
and call \(\bigl(c, f(c)\bigr)\) the point of tangency.
This slope and the instantaneous rate-of-change of \(f\)
can only be defined if \(f\) is continuous around \(c,\) if that limit exists.
When this limits exists, we say \(f\) is differentiable at \(x=c.\)
Big idea: rather than describing the instantaneous rate-of-change of \(f\) at a specific point \(x=c,\) we can keep \(x\) generic and define the function that describes the instantaneous rate-of-change of \(f\) at any point. The derivative of \(f\) with respect to \(x,\) denoted either as \(f'\) (which we read as “\(f\) prime”) or as \(\frac{\mathrm{d}f}{\mathrm{d}x},\) is \[ f'(x) = \lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}\,. \] The \(\frac{\mathrm{d}f}{\mathrm{d}x}\) notation alludes to the derivative being a slope; the \(\mathrm{d}\) in that notation may be read as “a small deviation in,” and so \(\frac{\mathrm{d}f}{\mathrm{d}x}\) may be read as “a small deviation in the output per a small deviation in the input.”