Source: AP Calculus BC, Section I (Parts A and B)
Difficulty: Intermediate to Advanced | Prerequisites: solid command of AB-level differentiation (power rule, product/quotient rule, basic trig derivatives) and a working understanding of limits. If the chain rule still feels shaky, revisit that first.
Derivatives, continuity, and limits form roughly a third of every AP Calculus BC exam. They are the engine behind everything else in the course: integration is built on antiderivatives, series convergence tests lean on limit behaviour, and differential equations are statements about derivatives. If you have missed recent classes, this is the material to rebuild on, because almost every later topic assumes you can differentiate fluently and spot continuity or differentiability issues on sight.
This set of notes draws on questions from the 2012 AP Calculus BC practice exam (both the no-calculator and calculator sections) and distils the derivative, continuity, and limit concepts they test. You should already be comfortable with basic differentiation rules and the idea of a limit before working through these.
Know the chain rule cold, especially with trig and inverse trig functions. Understand the precise relationship between continuity and differentiability (continuous does not guarantee differentiable). Be able to read a graph of f' and extract information about f, including relative extrema, concavity, and inflection points.
Chain rule
The derivative of a composite function f(g(x)) equals f'(g(x)) multiplied by g'(x). In simple terms, you differentiate the outer function and then multiply by the derivative of what is inside it.
Implicit differentiation
A technique for finding dy/dx when y is not isolated on one side of the equation. You differentiate both sides with respect to x, treating y as a function of x (so every time you differentiate a y-term, you tack on a dy/dx), then solve for dy/dx. Think of it as: differentiate everything in sight, then untangle.
Continuity at a point
A function f is continuous at x = c when three conditions hold: f(c) is defined, the limit of f(x) as x approaches c exists, and that limit equals f(c). In simple terms, you can draw through the point without lifting your pencil.
Differentiability
A function is differentiable at a point if its derivative exists there, which requires the function to be continuous and to have no sharp corner, cusp, or vertical tangent at that point. Think of it as: the curve is smooth enough to have a single well-defined tangent line.
L'Hopital's rule
When a limit gives an indeterminate form (0/0 or infinity/infinity), you may take the derivative of the numerator and the derivative of the denominator separately, then re-evaluate the limit. In simple terms, if plugging in gives 0/0, differentiate top and bottom and try again.
Critical point
A value of x where f'(x) = 0 or f'(x) is undefined (and f(x) exists). These are the candidates for local maxima or minima. Think of it as: places where the slope is flat or the derivative breaks.
Inflection point
A point where the concavity of f changes, from concave up to concave down or vice versa. This corresponds to where f''(x) changes sign. In simple terms, the curve switches from bending one way to bending the other.
Concavity
A function is concave up where f''(x) > 0 (the curve opens upward, like a bowl) and concave down where f''(x) < 0 (the curve opens downward, like a hill). On the graph of f', concave up for f means f' is increasing.
Horizontal asymptote
A horizontal line y = L that the graph of f approaches as x goes to positive or negative infinity. Found by evaluating the limit of f(x) as x approaches infinity. In simple terms, it is the value the function settles towards far out along the x-axis.
The derivative of sin^n(x) requires the chain rule: bring down the power, reduce it by one, then multiply by the derivative of the inner function.
d/dx [sin^3(x)] = 3 sin^2(x) cos(x). The outer function is ( )^3, the inner is sin(x).
This pattern applies to cos^n(x), tan^n(x), and any composed trig expression.
When differentiating compositions like e^(arcsin x), apply the chain rule layer by layer: derivative of e^u is e^u, then multiply by derivative of arcsin(x) which is 1/sqrt(1 - x^2).
Used when you cannot easily solve for y. Differentiate every term with respect to x, attaching dy/dx to any term containing y, then isolate dy/dx.
Example: if arcsin(x) = ln(y), then 1/sqrt(1 - x^2) = (1/y)(dy/dx), so dy/dx = y / sqrt(1 - x^2).
The exam tests whether you remember to apply the chain rule to y-terms. Every y you differentiate picks up a dy/dx factor.
Differentiability implies continuity, but continuity does not imply differentiability.
f(x) = sqrt(|x - 2|) is continuous at x = 2 (the limit exists and equals f(2) = 0), but not differentiable there because the absolute value creates a sharp corner.
Sharp corners, cusps, and vertical tangents all break differentiability while preserving continuity.
When a function is defined as continuous at x = c, the only guaranteed fact is that f(c) = lim f(x) as x approaches c from both sides. You cannot assume the derivative exists.
Where f'(x) > 0, f is increasing. Where f'(x) < 0, f is decreasing.
Where f' changes from positive to negative, f has a local maximum. Where f' changes from negative to positive, f has a local minimum.
Where f' is increasing, f is concave up. Where f' is decreasing, f is concave down.
Inflection points of f occur where f' changes from increasing to decreasing (or vice versa), which is where f'' changes sign.
When h(x) is defined as the integral from 0 to x of f(t) dt, then h'(x) = f(x) and h''(x) = f'(x) by the Fundamental Theorem of Calculus. So the graph of f gives you h' directly.
Compare the degrees of numerator and denominator.
Same degree: the horizontal asymptote is the ratio of leading coefficients.
Numerator degree lower: asymptote is y = 0.
Numerator degree higher: no horizontal asymptote.
For rational functions with polynomials in both numerator and denominator, divide top and bottom by the highest power of x to find the limit as x approaches infinity.
Example: y = (20x^2 - x)/(1 + 4x^2) has horizontal asymptote y = 20/4 = 5.
Applies only when direct substitution gives 0/0 or infinity/infinity.
Differentiate the numerator and denominator independently (do not use the quotient rule on the whole fraction).
Example: lim as x approaches 1 of [integral from 1 to x of g(t) dt] / [g(x) - 6]. If g(1) = 6, the numerator is 0 and denominator is 0. By L'Hopital: numerator derivative is g(x), denominator derivative is g'(x). Evaluate at x = 1 to get g(1)/g'(1) = 6/3 = 2.
To find points where the tangent has a specific slope, set f'(x) equal to that slope and solve.
For f(x) = x/(x + 2), the quotient rule gives f'(x) = 2/(x + 2)^2. Setting this equal to 1/2 gives (x + 2)^2 = 4, so x = 0 or x = -4. Both yield valid points on the graph.
When asked on what intervals one function's rate of change exceeds another's, compare their derivatives.
f(x) = e^x and g(x) = x^4: compare e^x with 4x^3. A graphing calculator shows they intersect near x = 0.831 and x = 7.384, with e^x < 4x^3 between those values.
Given a differential equation dy/dx = expression and points on the curve, you can find d^2y/dx^2 by differentiating the right side (using implicit differentiation if y appears).
If dy/dx = x^2 + y, then d^2y/dx^2 = 2x + dy/dx = 2x + x^2 + y. Evaluate at a point to determine concavity, which tells you whether it is a local max, min, or inflection point.
If f'(x) is increasing for x < 0 and decreasing for x > 0, then f' has a maximum at x = 0, meaning f'' changes from positive to negative there. So f has an inflection point at x = 0 (concavity changes), and f itself has a point where it is steepest.
The graph of f in this case could look like a function that is concave up on the left and concave down on the right.
\frac{d}{dx}[\sin^n(x)] = n\sin^{n-1}(x)\cos(x)\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v\,u' - u\,v'}{v^2}Continuity at x = c requires all three: f(c) exists, lim f(x) as x approaches c exists, and lim f(x) = f(c).
L'Hopital's rule: if lim f(x)/g(x) gives 0/0 or infinity/infinity, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.
Derivatives model rates of change in every applied field. The chain rule, for instance, is exactly the maths behind computing how fuel consumption changes with time when speed itself is changing (as in Q85 on this exam, where dF/dt = F'(s) times ds/dt). Engineers use implicit differentiation when design equations link variables that cannot be separated cleanly.
Students often think continuity and differentiability are the same thing. They are not. A function with a sharp corner (like sqrt(|x - 2|) at x = 2) is continuous but not differentiable.
Students sometimes apply L'Hopital's rule when the limit is not indeterminate. You must verify the 0/0 or infinity/infinity form before differentiating numerator and denominator.
When reading a graph of f', students frequently confuse "f' is positive" with "f is concave up". Positive f' means f is increasing. For concavity, you need f' to be increasing (not just positive).
Students often forget the chain rule's inner derivative. For d/dx[sin^3(x)], writing 3 sin^2(x) and stopping there is one of the most common errors on the exam.
Chain rule with trig functions appears on nearly every BC exam. Expect at least one question requiring d/dx of sin^n(x) or cos^n(x).
Graph-of-f' questions are a staple. You will be given a graph of the derivative and asked about the original function's maxima, minima, concavity, and inflection points.
Continuity vs differentiability is tested almost every year. The examiners like absolute value and piecewise functions for this.
L'Hopital's rule questions often hide the indeterminate form inside an integral expression (as in Q28 on this exam). Recognise the 0/0 setup before applying.
The Fundamental Theorem of Calculus linking h(x) = integral of f(t) dt with h' = f and h'' = f' is heavily tested in the multiple-choice section.
True or false: if f is continuous at x = a, then f is differentiable at x = a.
Fill in the blank: d/dx [sin^3(x)] = 3 sin^2(x) times ____.
True or false: L'Hopital's rule can be applied whenever a limit involves a fraction.
Fill in the blank: if f'(x) > 0 on an interval, then f is ____ on that interval.
True or false: an inflection point of f occurs where f''(x) = 0.
Answers: 1. False (continuity does not guarantee differentiability). 2. cos(x). 3. False (only when the form is indeterminate, 0/0 or infinity/infinity). 4. Increasing. 5. False (f'' must change sign, not merely equal zero).
Q: If y = sin^3(x), what is dy/dx?
A: 3 sin^2(x) cos(x). Apply the chain rule: the power rule gives 3 sin^2(x), then multiply by the derivative of sin(x), which is cos(x). Answer: (E).
Q: The function f(x) = sqrt(|x - 2|) is defined for all x. Which statement about f at x = 2 is true?
A: f is continuous but not differentiable at x = 2. The function equals zero at x = 2 and the limit exists, so continuity holds. But the absolute value creates a cusp, so no derivative exists there. Answer: (A).
Q: If arcsin(x) = ln(y), what is dy/dx?
A: y / sqrt(1 - x^2). Differentiate both sides implicitly: 1/sqrt(1 - x^2) = (1/y)(dy/dx). Multiply both sides by y. Answer: (A).
Q: The line y = 5 is a horizontal asymptote of which function?
A: y = (20x^2 - x) / (1 + 4x^2). As x approaches infinity, the ratio of leading coefficients is 20/4 = 5. Answer: (E).
Q: Let g be continuously differentiable with g(1) = 6 and g'(1) = 3. What is lim as x approaches 1 of [integral from 1 to x of g(t) dt] / [g(x) - 6]?
A: 2. Both numerator and denominator approach 0 as x approaches 1. By L'Hopital's rule, the limit becomes g(x) / g'(x) evaluated at x = 1, giving 6/3 = 2. Answer: (D).
Q: f(x) = x / (x + 2). At which points does the tangent line have slope 1/2?
A: (0, 0) and (-4, 2). The derivative is f'(x) = 2 / (x + 2)^2. Setting this equal to 1/2 gives (x + 2)^2 = 4, so x = 0 or x = -4. Answer: (C).
Q: If the function f is continuous at x = 3, which of the following must be true?
A: f(3) = lim (x->3-) f(x) = lim (x->3+) f(x). This is the definition of continuity. The derivative need not exist. Answer: (C).
The chain rule reappears throughout integration (reverse chain rule is u-substitution) and in series (differentiating power series term by term). Continuity and differentiability feed directly into the hypotheses of the Mean Value Theorem and the Fundamental Theorem of Calculus, both of which are tested in their own right. Graph analysis of f' is the mirror image of accumulation-function problems, where you integrate f to get a new function.
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