Latest · Derivatives
The Chain Rule: How It Works, With Worked Examples
Learn the chain rule step by step: what it says, why it works, how to spot the inside function, and worked examples with powers, trig, e^x and ln.
· 5 min read
Calculus, explained step by step
Clear explanations, graphs and worked examples for every topic from limits to multivariable calculus. Read the idea, then practice it right away.
Latest posts
Continuity in Calculus: Definition, Discontinuities and the IVT
Learn the three conditions for continuity, how to classify removable, jump and infinite discontinuities, and how to use the Intermediate Value Theorem.
· 7 min read
Antiderivatives: Reverse Power Rule, +C and Initial Value Problems
Learn how to find antiderivatives: the reverse power rule, a table of basic integrals, why you add +C, and how to solve initial value problems step by step.
· 7 min read
AP Calculus AB vs BC: Differences, Exam Format and Which to Take
AP Calculus AB vs BC explained: the extra topics in BC, how the exams are structured, how hard each one is, and how to decide which course is right for you.
· 7 min read
Curve Sketching With Derivatives: First and Second Derivative Tests
Sketch any curve with derivatives: increasing and decreasing intervals, local extrema, concavity, inflection points and a full cubic worked example.
· 7 min read
Improper Integrals: Convergence, p-Integrals and Examples
Learn how to evaluate improper integrals with infinite limits or infinite discontinuities, the p-integral rule, the comparison test and worked examples.
· 7 min read
Series Convergence Tests: Which Test to Use and When
Every series convergence test with examples: nth-term, integral, p-series, comparison, ratio, root and alternating, plus a which-test table.
· 7 min read
Limits and Continuity
What limits mean, how to evaluate them, and how continuity connects them to the rest of calculus.
How to Evaluate Limits: 5 Methods With Worked Examples
Evaluate limits step by step: direct substitution, factor and cancel, rationalizing, special trig limits, and a flowchart for picking a method.
· 7 min read
L'Hôpital's Rule: When and How to Use It, With Examples
Learn L'Hôpital's rule for 0/0 and ∞/∞ limits, how to handle 0·∞, ∞ − ∞ and 1^∞ forms, worked examples, and when the rule does not apply.
· 7 min read
Limits at Infinity: End Behavior and Horizontal Asymptotes
Learn how to find limits at infinity: the degree rules for rational functions, horizontal asymptotes, square roots, and worked examples with graphs.
· 8 min read
Derivatives
Derivative rules explained step by step, from the power rule to implicit differentiation.
Derivatives of Exponential and Log Functions: e^x, a^x and ln x
Learn the derivatives of e^x, a^x, ln x and log base a, why e is special, and how logarithmic differentiation handles x^x and messy products.
· 6 min read
Derivatives of Trig Functions: All Six, With Proofs and Examples
The derivatives of sin, cos, tan, cot, sec and csc, why the derivative of sin x is cos x, and worked chain rule examples with trig functions.
· 7 min read
Implicit Differentiation: Step-by-Step Method and Examples
Learn implicit differentiation step by step: find dy/dx for circles and curves you cannot solve for y, find tangent lines, and compute the second derivative.
· 6 min read
Applications of Derivatives
Optimization, related rates, curve sketching and linear approximation, with worked examples.
Linear Approximation: Tangent Line Estimates and Differentials
Learn linear approximation step by step: the tangent line formula, estimating square roots like sqrt(4.1), differentials, error and over or underestimates.
· 7 min read
Mean Value Theorem: Statement, Rolle's Theorem and Finding c
Understand the mean value theorem with graphs: what it says, Rolle's theorem, how to find the value of c step by step, and when the theorem does not apply.
· 7 min read
Optimization Problems in Calculus: A 5-Step Method With Examples
Solve calculus optimization problems with a reliable 5-step method: box, fence and can examples worked in full, plus how to check endpoints.
· 7 min read
Integrals
Antiderivatives, Riemann sums, definite integrals and the Fundamental Theorem of Calculus.
Area Between Curves: Top Minus Bottom, Step by Step
Find the area between two curves step by step: top minus bottom, finding intersection points, curves that cross, and when to integrate with respect to y.
· 6 min read
Fundamental Theorem of Calculus: Parts 1 and 2 With Examples
Learn both parts of the fundamental theorem of calculus: evaluate definite integrals with antiderivatives and differentiate integrals with variable limits.
· 7 min read
Riemann Sums: Left, Right and Midpoint Rules With Examples
Learn Riemann sums step by step: left, right and midpoint rectangles, sigma notation, over and underestimates, and how the limit becomes the definite integral.
· 7 min read
Integration Techniques
Substitution, integration by parts, partial fractions, trig integrals and improper integrals.
Integration by Parts: Formula, LIATE and Worked Examples
Master integration by parts: the formula, how LIATE picks u, the tabular shortcut, the looping e^x sin x trick, and fully worked examples.
· 7 min read
Partial Fractions: How to Decompose and Integrate, Step by Step
Learn partial fraction decomposition for distinct linear, repeated linear and irreducible quadratic factors, the cover-up shortcut, and worked integrals.
· 7 min read
Trig Substitution: The Three Cases, Triangles and Examples
Learn trigonometric substitution: which of the three substitutions to use, how to build the reference triangle, and how to convert the answer back to x.
· 7 min read
Sequences and Series
Convergence tests, power series and Taylor series, explained with examples you can follow.
Geometric Series: Sum Formula, Convergence and Examples
Learn the geometric series sum formula, why it converges only when |r| < 1, how partial sums behave, and how to turn repeating decimals into fractions.
· 7 min read
Power Series: Radius and Interval of Convergence, With Examples
Find the radius and interval of convergence of a power series with the ratio test, check both endpoints correctly, and follow five fully worked examples.
· 8 min read
Taylor Series and Maclaurin Series: Formula, Examples, Error Bound
Learn the Taylor and Maclaurin series formulas, memorize the big four series, build new series from old ones and bound the error of a Taylor polynomial.
· 7 min read
Multivariable Calculus
Vectors, partial derivatives, gradients and multiple integrals for Calculus 3.
Double Integrals: Iterated Integrals, Regions and Polar Form
Evaluate double integrals step by step: iterated integrals, Fubini's theorem, curved regions, switching the order and polar coordinates.
· 7 min read
Gradient and Directional Derivatives: Formula and Examples
Learn what the gradient vector is, how to compute a directional derivative with a unit vector, and why the gradient points in the direction of steepest ascent.
· 7 min read
Lagrange Multipliers: Method, Intuition and Worked Examples
Learn the method of Lagrange multipliers for constrained optimization, why the gradients must be parallel, and how to solve the equations in worked examples.
· 8 min read
Study Tips and Exam Prep
How to study calculus, avoid common mistakes and walk into your exam prepared.
Calculus 1 vs 2 vs 3: What Each Course Covers and Which Is Hardest
Compare Calculus 1, 2 and 3: topics in each course, prerequisites, difficulty, which one is hardest and why, plus a side-by-side comparison table.
· 8 min read
12 Common Calculus Mistakes (and How to Fix Each One)
The 12 calculus mistakes that cost students the most points, from the chain rule factor to +C and sign errors, each shown wrong vs right with worked math.
· 8 min read
How to Prepare for a Calculus Exam: A 2-Week Study Plan
A day-by-day 2-week plan for your calculus exam: timed practice, a smart formula sheet, exam-day tactics and fast ways to check your answers.
· 7 min read
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