Overview
The derivative measures instantaneous rate of change. Geometrically, it is the slope of the tangent line. In motion problems, the derivative of a position function represents velocity (velocity is the slope of a position-time graph).
Limit definition of the derivative
There are two limit definitions of the derivative, one is a limit as \(h\to 0\) and the other is a limit as \(x\to a\):\[ f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h} \;=\; \lim_{x\to a}\frac{f(x)-f(a)}{x-a}. \]
Example
Let \(\displaystyle f(x)=x^3-2x\). Compute \(f'(1)\):
\begin{align*} \lim_{h\to 0} \frac{f(1+h)-f(1)}{h}&= \lim_{h\to 0} \frac{\big[(1+h)^3-2(1+h)\big]-\big[1-2\big]}{h}\\ &=\lim_{h\to 0} \frac{\big[(1+3h+3h^2+h^3) - 2-2h\big]-\big[-1\big]}{h}\\ &=\lim_{h\to 0} \frac{h+3h^2+h^3}{h}\\ &=\lim_{h\to 0} 1+3h+h^2 \\ &= 1 \end{align*}
So \(f'(1)=1\). Tangent slope at \(x=1\) is \(1\).
Tangent line at a point
- Find the slope \(m=f'(a)\).
- Use point–slope form with the point \((a,f(a))\): \(y-f(a)=m(x-a)\).
Example: Tangent to a polynomial
Let \(y=x^3-2x\) at \(x=1\). We found \(f'(1)=1\) and \(f(1)=-1\). Equation: \(y+1=1(x-1)\Rightarrow y=x-2.\)
Example: Tangent line given it passes through another point
The tangent to \(y=f(x)\) at \((4,-1)\) passes through \((1,5)\). Then the tangent slope is \(\displaystyle m=\frac{-1-5}{4-1}=-2\). So \(f(4)=-1\) and \(f'(4)=-2\).
Recognize “this limit is \(f'(a)\)”
Example: Identify \(f\) and \(a\)
\(\displaystyle \lim_{h\to0}\frac{(2+h)^4-2^4}{h}\) is \(f'(a)\) for \(f(x)=x^4\) at \(a=2\).
Example: Another form
\(\displaystyle \lim_{x\to 3}\frac{\sqrt{x}-\sqrt{3}}{x-3}\) matches \(\frac{f(x)-f(3)}{x-3}\) with \(f(x)=\sqrt{x}\).
Estimate derivatives from tables & graphs
Example: From a table
Given \(s(t)\) (meters) at \(t\) (seconds): \(s(1.9)=4.82,\;s(2.0)=5.00,\;s(2.1)=5.23\).
Estimate \(s'(2)\approx\displaystyle \frac{s(2.1)-s(1.9)}{0.2}=\frac{5.23-4.82}{0.2}=2.05\) m/s.
Example: From a graph
Calculate two secant slopes straddling the \(x\)-value and average them to approximate the tangent slope.
Position & velocity
If \(s(t)\) is position, then \(v(t)=s'(t)\). On a position–time graph:
- \(v(t)>0\): moving to the right (increasing position).
- \(v(t)<0\): moving to the left (decreasing position).
- \(v(t)=0\): stopped (horizontal tangent).
Example: Classifying intervals
Suppose the position graph rises on \([0,1)\), is flat on \([1,2]\), falls on \((2,3)\), and rises again on \((3,4]\). Then the particle moves right on \([0,1)\cup(3,4]\), is still on \([1,2]\), and left on \((2,3)\).