Home Integration by PartsIntegration by PartsExamples Integration by Parts with a definite integral Going in Circles Tricks of the Trade Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric FunctionsProduct of Sines and Cosines (mixed even and odd powers or only odd powers) Product of Sines and Cosines (only even powers) Product of Secants and Tangents Other Cases Trig SubstitutionsHow Trig Substitution WorksSummary of trig substitution options Examples Completing the Square Partial FractionsIntroduction to Partial FractionsLinear Factors Irreducible Quadratic Factors Improper Rational Functions and Long Division Summary Strategies of IntegrationSubstitutionIntegration by Parts Trig Integrals Trig Substitutions Partial Fractions Improper IntegralsType 1  Improper Integrals with Infinite Intervals of IntegrationType 2  Improper Integrals with Discontinuous Integrands Comparison Tests for Convergence Modeling with Differential EquationsIntroductionSeparable Equations A Second Order Problem Euler's Method and Direction FieldsEuler's Method (follow your nose)Direction Fields Euler's method revisited Separable EquationsThe Simplest Differential EquationsSeparable differential equations Mixing and Dilution Models of GrowthExponential Growth and DecayThe Zombie Apocalypse (Logistic Growth) Linear EquationsLinear ODEs: Working an ExampleThe Solution in General Saving for Retirement Parametrized CurvesThree kinds of functions, three kinds of curvesThe Cycloid Visualizing Parametrized Curves Tracing Circles and Ellipses Lissajous Figures Calculus with Parametrized CurvesVideo: Slope and AreaVideo: Arclength and Surface Area Summary and Simplifications Higher Derivatives Polar CoordinatesDefinitions of Polar CoordinatesGraphing polar functions Video: Computing Slopes of Tangent Lines Areas and Lengths of Polar CurvesArea Inside a Polar CurveArea Between Polar Curves Arc Length of Polar Curves Conic sectionsSlicing a ConeEllipses Hyperbolas Parabolas and Directrices Shifting the Center by Completing the Square Conic Sections in Polar CoordinatesFoci and DirectricesVisualizing Eccentricity Astronomy and Equations in Polar Coordinates Infinite SequencesApproximate Versus Exact AnswersExamples of Infinite Sequences Limit Laws for Sequences Theorems for and Examples of Computing Limits of Sequences Monotonic Covergence Infinite SeriesIntroductionGeometric Series Limit Laws for Series Test for Divergence and Other Theorems Telescoping Sums Integral TestPreview of Coming AttractionsThe Integral Test Estimates for the Value of the Series Comparison TestsThe Basic Comparison TestThe Limit Comparison Test Convergence of Series with Negative TermsIntroduction, Alternating Series,and the AS TestAbsolute Convergence Rearrangements The Ratio and Root TestsThe Ratio TestThe Root Test Examples Strategies for testing SeriesStrategy to Test Series and a Review of TestsExamples, Part 1 Examples, Part 2 Power SeriesRadius and Interval of ConvergenceFinding the Interval of Convergence Power Series Centered at $x=a$ Representing Functions as Power SeriesFunctions as Power SeriesDerivatives and Integrals of Power Series Applications and Examples Taylor and Maclaurin SeriesThe Formula for Taylor SeriesTaylor Series for Common Functions Adding, Multiplying, and Dividing Power Series Miscellaneous Useful Facts Applications of Taylor PolynomialsTaylor PolynomialsWhen Functions Are Equal to Their Taylor Series When a Function Does Not Equal Its Taylor Series Other Uses of Taylor Polynomials Functions of 2 and 3 variablesFunctions of several variablesLimits and continuity Partial DerivativesOne variable at a time (yet again)Definitions and Examples An Example from DNA Geometry of partial derivatives Higher Derivatives Differentials and Taylor Expansions Differentiability and the Chain RuleDifferentiabilityThe First Case of the Chain Rule Chain Rule, General Case Video: Worked problems Multiple IntegralsGeneral Setup and Review of 1D IntegralsWhat is a Double Integral? Volumes as Double Integrals Iterated Integrals over RectanglesHow To Compute Iterated IntegralsExamples of Iterated Integrals Fubini's Theorem Summary and an Important Example Double Integrals over General RegionsType I and Type II regionsExamples 14 Examples 57 Swapping the Order of Integration Area and Volume Revisited Double integrals in polar coordinatesdA = r dr (d theta)Examples Multiple integrals in physicsDouble integrals in physicsTriple integrals in physics Integrals in Probability and StatisticsSingle integrals in probabilityDouble integrals in probability Change of VariablesReview: Change of variables in 1 dimensionMappings in 2 dimensions Jacobians Examples Bonus: Cylindrical and spherical coordinates 
The General Version of the Chain Rule starts with a function $f(x,y)$, where $x$ and $y$ are themselves functions $x = x(s,\, t)$ and $y = y(s,\,t)$ of two other variables $s$ and $ t$, so that the composition
$${\color{darkerblue}z\ = \ f(x(s, \,t), y(s, \,t))}$$
is now a function of $s$ and $ t$. The partial derivatives of $z$ become:
Let's see why these formulas work. The partial derivative $\partial z/\partial s$ means ``Hold $t$ fixed and treat $z$ as a (compound) function of a single variable $s$. Then take its derivative.'' But this means that $x$ and $y$ are also treated as functions of the single variable $s$, and we are back in the setting of the simple case of the chain rule. The derivative of $z=f(x,y)$ is $f_x$ times the derivative of $x$ plus $f_y$ times the derivative of $y$, which is precisely what our first equation is saying. The reasoning behind the second equation is similar. The one and two variable chain rules set the pattern for more variables. If $w = f(x, \,y,\, z)$ and $$x \ = \ x(r, \,s,\, t), \qquad y \ = \ y(r, \,s, \,t), \qquad z \ = \ z(r,\, s, \,t),$$ then $$w \ = \ f(x(r, \,s, \,t), y(r,\, s, \,t), z(r, \,s, \,t))$$ is a function of $r, \,s,$ and $ t$ such that $$\frac{\partial w}{\partial r} \ = \ \frac{\partial f}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial r} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial r}\,$$ and so on for functions $f(x_1,\, x_2,\, \ldots, \, x_n)$ of $n$ variables for any $n$.
