The CalculusDX blog
CalculusDX, explained clearly
Plain-language explanations of the ideas in a first calculus course, from limits to integrals. Every article answers the question up front, then works through examples step by step.
Limits
What is a limit in calculus? A plain-language explanation
A limit is the value a function's outputs approach as the input gets close to a number. See what that means, how to read the notation, and worked examples.
7 min readLimitsHow to evaluate limits: every method, step by step
Learn how to evaluate limits: plug in first, then factor, use the conjugate, combine fractions, or check signs. Worked examples for every case.
9 min readLimitsLimits at infinity and horizontal asymptotes, explained
How to find limits at infinity and horizontal asymptotes: the degree rule for fractions, square roots, and differences, with worked examples.
8 min readLimitsContinuity and the Intermediate Value Theorem, explained simply
What it means for a function to be continuous, the three conditions to check, the kinds of discontinuity, and how to use the Intermediate Value Theorem.
8 min readDerivatives
What is a derivative? Slope, rate of change, and the definition
A derivative is the instantaneous rate of change of a function: the slope of its tangent line. See the limit definition, notation, and worked examples.
8 min readDerivativesDerivative rules: the complete list, with examples
Every derivative rule in CalculusDX in one place: power, product, quotient, chain, exponential, log and trig rules, with worked examples and how to choose.
8 min readDerivativesThe product rule and quotient rule: how and when to use each
The product rule (fg)'=f'g+fg' and the quotient rule (f/g)'=(f'g-fg')/g², why they work, step-by-step examples, and when a rewrite is quicker.
7 min readDerivativesThe chain rule explained, with step-by-step examples
The chain rule differentiates a function inside another function: outside derivative times inside derivative. Learn how to spot it and use it, step by step.
7 min readDerivativesImplicit differentiation, step by step
How to find dy/dx when an equation isn't solved for y: differentiate both sides, use the chain rule on y, and solve. Worked examples and common mistakes.
7 min readDerivativesDerivatives of eˣ, ln x, and other exponential and log functions
The derivative of e^x is e^x and the derivative of ln x is 1/x. Learn the rules for b^x and log base b, the chain rule versions and logarithmic differentiation.
7 min readApplications
How to find absolute maximum and minimum values
Find the absolute max and min of a function on a closed interval: find the critical numbers, then compare the function's values there and at the endpoints.
7 min readApplicationsThe Mean Value Theorem in plain language
The Mean Value Theorem says that somewhere, the instant rate of change equals the average rate. See what it means, how to find c, and what it's used for.
7 min readApplicationsL'Hôpital's rule: how it works and when to use it
L'Hôpital's rule evaluates 0/0 and ∞/∞ limits by differentiating the top and bottom separately. Learn when it applies, the steps, and the mistakes to avoid.
7 min readApplicationsRelated rates: a step-by-step method with worked examples
Solve related rates problems with one six-step method: picture, name, relate, differentiate, substitute, interpret. Ladder, balloon, cone and car examples.
8 min readApplicationsIncreasing, decreasing, and concavity: the first and second derivative tests
Use f′ to find where a function increases or decreases and its local max and min, and f″ to find concavity and inflection points. Sign charts, worked examples.
8 min readApplicationsHow to solve optimization problems in calculus
A step-by-step method for calculus optimization problems: write the objective, use the constraint, find the domain, compare critical numbers. Worked examples.
8 min readIntegrals
What is an integral? Area, Riemann sums, and the definite integral
An integral adds up infinitely many tiny pieces: area under a curve, distance from speed. See Riemann sums, the definite integral, net area and notation.
7 min readIntegralsThe Fundamental Theorem of Calculus, explained
The Fundamental Theorem of Calculus links derivatives and integrals. See both parts in plain language, why they're true, and worked examples of each.
8 min readIntegralsU-substitution: how to choose u, with step-by-step examples
U-substitution reverses the chain rule to solve integrals. Learn how to choose u, find du, change the limits of definite integrals, and avoid common mistakes.
6 min read