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Rate of Change
at a Point

Master the transition from average rate of change to instantaneous rate of change: the foundation of the derivative.

AP Calculus BC

What Is Rate of Change?

Rate of change measures how quickly one quantity changes relative to another. If you drive 120 miles in 2 hours, your average speed is \(\frac{120}{2} = 60\) mph: that is a rate of change.

Average Rate of Change: The Secant Slope

The average rate of change of \(f\) on the interval \([a,b]\) is the slope of the secant line through \((a, f(a))\) and \((b, f(b))\):

This is identical to the familiar slope formula \(\frac{\Delta y}{\Delta x}\). It tells you the overall rate across the entire interval, not what happens at any single point.

Visualizing the Secant Line

x y a b secant line f(x) Δx Δy

From Average to Instantaneous

What if you want the rate of change at a single moment, not over an interval? Shrink the interval: let \(b\) slide closer and closer to \(a\).

a tangent b₃ b₂ b₁

The Limit Definition

The instantaneous rate of change of \(f\) at \(x = a\) is defined as the limit of the average rate of change as the interval width shrinks to zero:

Equivalently, using two-point form:

Both expressions say the same thing: take the slope of the secant, then let the two points collapse to one.

Geometric Meaning: Tangent Slope

x y (a, f(a)) tangent at a f(x)

Average vs. Instantaneous

Average Rate of Change
Slope of the secant line between two points. Uses a finite interval \([a, b]\). No limit required: just algebra.
Instantaneous Rate of Change
Slope of the tangent line at a single point. Requires a limit as \(h \to 0\). This is the derivative \(f'(a)\).

Side-by-Side Comparison

Average ROC Instantaneous ROC
Formula \(\frac{f(b)-f(a)}{b-a}\) \(\displaystyle\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\)
Geometry Secant line slope Tangent line slope
Interval Finite: \([a, b]\) Single point: \(x = a\)
Requires limit? No Yes
Other name Difference quotient Derivative at a point

How to Compute \(f'(a)\)

1
Write the difference quotient. Form \(\frac{f(a+h) - f(a)}{h}\) by substituting \(a + h\) into \(f\).
2
Expand and simplify. Algebraically simplify the numerator so you can cancel the \(h\) in the denominator.
3
Cancel \(h\). Factor out and divide by \(h\). The expression should now be well-defined at \(h = 0\).
4
Take the limit. Evaluate \(\lim_{h \to 0}\) by direct substitution. The result is \(f'(a)\).

Example 1: \(f(x) = x^2\) at \(x = 3\)

1
Difference quotient: \(\frac{(3+h)^2 - 3^2}{h}\)
2
Expand: \(\frac{9 + 6h + h^2 - 9}{h} = \frac{6h + h^2}{h}\)
3
Cancel \(h\): \(\frac{h(6+h)}{h} = 6 + h\)
4
Limit: \(\lim_{h \to 0}(6+h) = \) \(6\)

Example 2: \(f(x) = \frac{1}{x}\) at \(x = 2\)

1
Difference quotient: \(\displaystyle\frac{\frac{1}{2+h} - \frac{1}{2}}{h}\)
2
Combine fractions in numerator: \(\displaystyle\frac{\frac{2 - (2+h)}{2(2+h)}}{h} = \frac{-h}{2h(2+h)}\)
3
Cancel \(h\): \(\displaystyle\frac{-1}{2(2+h)}\)
4
Limit: \(\displaystyle\lim_{h \to 0}\frac{-1}{2(2+h)} = \) \(-\frac{1}{4}\)

Example 3: \(f(x) = \sqrt{x}\) at \(x = 9\)

Step 1: \(\displaystyle\frac{\sqrt{9+h} - 3}{h}\). Direct substitution gives \(\frac{0}{0}\), so we rationalize.

Step 2: Multiply by the conjugate and simplify the numerator: $$\frac{\sqrt{9+h}-3}{h}\cdot\frac{\sqrt{9+h}+3}{\sqrt{9+h}+3} = \frac{(9+h)-9}{h(\sqrt{9+h}+3)} = \frac{h}{h(\sqrt{9+h}+3)}$$

Step 3: Cancel \(h\) and take the limit: $$\lim_{h\to 0}\frac{1}{\sqrt{9+h}+3} = \frac{1}{3+3} = \frac{1}{6}$$

The Alternate Form of the Derivative

Instead of using \(h\), you can write the derivative using a second variable \(x\) approaching \(a\):

This is sometimes easier when you can factor the numerator. For instance, to find the derivative of \(f(x) = x^3\) at \(x = 2\):

Common Mistakes

✕ Forgetting the Limit
Writing \(\frac{f(a+h)-f(a)}{h}\) without \(\lim_{h\to 0}\) gives only the difference quotient, not the derivative.
✕ Plugging \(h=0\) Too Early
Substituting \(h=0\) before simplifying yields \(\frac{0}{0}\). You must cancel \(h\) first, then take the limit.
✕ Algebra Errors in Expansion
Expanding \((a+h)^2\) as \(a^2+h^2\) (missing the \(2ah\) term) is the most common source of wrong answers.

Question 1 of 5

The average rate of change of \(f(x) = x^2\) on the interval \([1, 4]\) is:

Question 2 of 5

Which expression represents the instantaneous rate of change of \(f\) at \(x = a\)?

Question 3 of 5

Using the limit definition, find \(f'(5)\) for \(f(x) = 3x - 7\).

Question 4 of 5

Geometrically, the instantaneous rate of change at a point on a curve is the slope of the:

Question 5 of 5

Find the instantaneous rate of change of \(f(x) = x^2 + 4x\) at \(x = -1\).