Difficulty: Introductory to Intermediate | Prerequisites: Chapter 1 (limits, continuity, slope of secant lines). You should be comfortable evaluating limits before starting here.
Chapter 2 is where calculus properly begins. In Chapter 1 you spent time on limits, working out what happens as x approaches some value. Now you put that machinery to work: the derivative is a limit that measures how fast a function is changing at a single point. This is the instantaneous rate of change, and it underpins everything from velocity in physics to marginal cost in economics. If you missed the limits material, go back to Chapter 1 first; every derivative calculation here rests on it.
The derivative of a function at a point is the slope of the tangent line there, computed as a limit. You can find it at a single number c, or as a whole new function f'(x). A function that has a derivative at a point is called differentiable there; cusps, discontinuities, and vertical tangents are the three ways differentiability fails.
Derivative of f at a number c
The number f'(c) = lim(h→0) [f(c+h) − f(c)] / h, provided the limit exists. An equivalent form is f'(c) = lim(x→c) [f(x) − f(c)] / (x − c). Think of it as the exact slope of the curve at one specific point.
Tangent line
The line through (c, f(c)) whose slope equals f'(c). In point-slope form: y − f(c) = f'(c)(x − c). In simple terms, the tangent line is the best straight-line approximation of the curve at that point.
Derivative function (f')
The function f'(x) = lim(h→0) [f(x+h) − f(x)] / h. Instead of finding the slope at one point, you leave x as a variable and get a formula that works everywhere the limit exists. Think of it as a machine: feed it any x value and it returns the slope there.
Leibniz notation
Writing df/dx or (d/dx)f(x) instead of f'(x). The derivative at x = c is written df/dx evaluated at x = c. In simple terms, df/dx reads as "the rate of change of f with respect to x."
Differentiable at c
A function f is differentiable at c if f'(c) exists. A function is differentiable on an interval (a,b) if it is differentiable at every number in that interval. Think of it as: the curve is smooth enough at that point to have a well-defined slope.
Not differentiable
A function fails to be differentiable at a point where the derivative does not exist. At such a point the function is called not differentiable. In simple terms, something goes wrong with the slope: a sharp corner, a break, or a vertical tangent.
Form 1 (x → c): f'(c) = lim(x→c) [f(x) − f(c)] / (x − c). You compare the function's value at a nearby point x to its value at c, then shrink the gap.
Form 2 (h → 0): f'(c) = lim(h→0) [f(c+h) − f(c)] / h. You nudge the input by a small amount h and see how the output changes per unit of h. This is the form used most often.
Both give the same number. Choose whichever makes the algebra cleaner for a given problem.
Once you have f'(c), the tangent line equation follows from point-slope form:
y − f(c) = f'(c)(x − c)
This is the line through (c, f(c)) with slope equal to the derivative.
To find f'(3) for f(x) = 2x + 5:
Set up the limit: lim(h→0) [f(3+h) − f(3)] / h = lim(h→0) [(2(3+h)+5) − (2·3+5)] / h
Simplify the numerator: (6+2h+5) − 11 = 2h
Cancel: 2h/h = 2
The limit is 2. This makes sense: f(x) = 2x + 5 is a line with slope 2.
For f(x) = √(x+1), finding f'(3) requires rationalising the numerator. Multiply top and bottom by √(3+h+1) + √(3+1) to clear the square roots, then cancel h.
The derivative f'(x) carries units of (units of f) / (units of x). If s is position in feet and t is time in seconds, then s'(t) is velocity in feet per second.
If the position of an object is s(t), its instantaneous velocity at time t is v(t) = s'(t). For example, if s = 4 + 30t − 16t², the velocity at t = 1 is found by computing s'(1) using the limit definition.
To approximate f'(c) from a graph, draw the tangent line at x = c and estimate its slope using two points on that line. Rise over run gives you the derivative.
When you leave x as a variable in the limit definition, f'(x) = lim(h→0) [f(x+h) − f(x)] / h, you get the derivative function. It returns the slope at any x where the limit exists.
Example: f(x) = x² + x − 8.
f(x+h) = (x+h)² + (x+h) − 8 = x² + 2xh + h² + x + h − 8
f(x+h) − f(x) = 2xh + h² + h
Divide by h: 2x + h + 1
Take the limit as h → 0: f'(x) = 2x + 1
Example: g(x) = 5/x.
g(x+h) = 5/(x+h)
g(x+h) − g(x) = 5/(x+h) − 5/x = [5x − 5(x+h)] / [x(x+h)] = −5h / [x(x+h)]
Divide by h: −5 / [x(x+h)]
Take the limit as h → 0: g'(x) = −5/x²
A function is differentiable on (a, b) if f'(c) exists for every c in that interval.
Cusp or corner (jagged edge): The left-hand and right-hand limits of the difference quotient disagree. Example: f(x) = |x − 1| at x = 1. The slope from the left is −1; the slope from the right is +1. The two-sided limit does not exist.
Discontinuity: If f is not continuous at c, then f is not differentiable at c. This follows from the theorem: if f is differentiable at c, then f is continuous at c. The contrapositive does the work. Example: a piecewise function with a jump at x = 3.
Vertical tangent: The tangent line is vertical, so the slope is undefined (the limit goes to ±∞). Example: g(x) = x^(1/3) at x = 0. The curve is smooth and continuous there, but the tangent line is straight up and down.
If f is differentiable at c, then f is continuous at c. The converse is false: f(x) = |x| is continuous at x = 0 but not differentiable there.
Three-step process:
Find where f'(x) = 0: these are the x-values where the tangent to f is horizontal (peaks, troughs, flat spots). Plot these as zeros of f'.
Determine where f is increasing (f' > 0) and where f is decreasing (f' < 0).
Estimate the actual slope at a few representative points in each interval to get the shape of f'.
f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}\text{Tangent line: } y - f(c) = f'(c)(x - c)f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}Algebra refresher for later sections: 1/x^n = x^(−n) and the nth root of x = x^(1/n).
The derivative measures how fast something is changing at a single instant. Velocity is the derivative of position: if you know s(t), then s'(t) tells you the speed and direction of travel right now. Economists use the same idea for marginal cost and marginal revenue, measuring the additional cost or revenue from one more unit. In biology, the derivative of a population function gives the growth rate at a given moment. Temperature change, infection spread, chemical reaction rates: all instantaneous rates of change, all derivatives.
Students often think that if a function is continuous at a point, it must be differentiable there. It does not follow. Continuity is necessary for differentiability, but it is not sufficient. The classic counterexample is f(x) = |x| at x = 0: continuous, but not differentiable.
Students frequently confuse the derivative at a point (a single number) with the derivative function (a formula). f'(3) is a number; f'(x) is a function.
When using the limit definition, a common error is to forget to subtract f(c) in the numerator, or to cancel h before taking the limit. If h is still in the denominator when you try to evaluate the limit, you have not simplified enough.
Some students think a vertical tangent means the function is discontinuous. It does not. The function can be perfectly continuous (like the cube root function at x = 0) and still have a vertical tangent.
⚠️ Computing derivatives from the limit definition is a major learning objective. Expect at least one exam question that requires the full limit process, not just applying shortcut rules.
⚠️ You will be asked to find the equation of a tangent line. Know the workflow: compute f'(c), then plug into point-slope form.
⚠️ Identifying where a function is not differentiable (from a graph or a formula) is a common exam topic. Be ready to name the reason: cusp, discontinuity, or vertical tangent.
⚠️ The theorem "differentiable implies continuous" and its false converse appear regularly on exams, often as true/false or short-answer questions.
True or false: If f is differentiable at x = 5, then f is continuous at x = 5.
True or false: If f is continuous at x = 5, then f is differentiable at x = 5.
Fill in the blank: The derivative f'(c) equals the slope of the ______ line to the graph of f at x = c.
True or false: f(x) = |x| is differentiable at x = 0.
Fill in the blank: The units of the derivative are the units of ______ divided by the units of ______.
Q: Use the limit definition to find f'(2) for f(x) = x² − 3x.
A: f'(2) = lim(h→0) [(2+h)² − 3(2+h) − (4−6)] / h = lim(h→0) [4+4h+h² − 6 − 3h + 2] / h = lim(h→0) [h+h²] / h = lim(h→0) (1+h) = 1.
Q: Find the equation of the tangent line to f(x) = √(5−x) at x = 1.
A: f(1) = 2. Using the limit definition, f'(1) = −1/4. The tangent line is y − 2 = (−1/4)(x − 1), which simplifies to y = (−1/4)x + 9/4.
Q: Give an example of a function that is continuous at x = 0 but not differentiable at x = 0, and explain why.
A: f(x) = |x|. It is continuous everywhere, but the left-hand slope at 0 is −1 and the right-hand slope is +1. The two-sided limit of the difference quotient does not exist, so f is not differentiable at 0.
Q: The height of a ball is given by h(t) = −16t² + 96t + 10. Use the limit definition to find the velocity function v(t), and determine whether the ball is moving up or down at t = 4.
A: v(t) = h'(t) = −32t + 96. At t = 4, v(4) = −32(4) + 96 = −32. Since v(4) < 0, the ball is moving downward.
Q: A function f has f'(x) = 5/(x+1)². What are the units of f'(x) if f is measured in metres and x in seconds?
A: Metres per second (m/s).
The limit definition here is the foundation for every differentiation rule in Sections 3 through 5 (power rule, product rule, quotient rule, chain rule). Those rules are shortcuts that spare you the limit process, but they are all proved from it. The concept of differentiability connects back to continuity from Chapter 1 and forward to implicit differentiation (Section 6) and related rates (Section 8), where derivatives are taken of equations that are not solved for y.
derivative, derivative definition, limit definition of derivative, difference quotient, tangent line, tangent line equation, point-slope form, instantaneous rate of change, average rate of change, slope of tangent, differentiable, not differentiable, cusp, corner, vertical tangent, continuous vs differentiable, derivative as a function, Leibniz notation, dy/dx, f prime, velocity as derivative, MTH 132, Calculus I, Michigan State University, Chapter 2