A salvation guide to Calculus - Make knowledge connected

Explore differentiation and integration as two sides of calculus, revealing how relationships change, enabling sine, Taylor series, Fourier analysis, and data science modeling through optimization.

See how differentiation and integration express the relationship between change and quantity, using velocity and distance as guiding examples. Build abstract symbolic methods and limits to obtain accurate, general relationships.

Learn how calculus models real world relationships by using six fundamental functions from acceleration, cyclical and natural change; apply differentiation, integration, and combination rules to tackle complexity.

Differentiate polynomial functions using the binomial theorem and delta x limits to obtain the derivative. Extend this method to differentiate the exponential function.

Explore differentiating exponential functions by converting to base e, showing that the derivative of a e^x is a e^x, with a representing the growth rate.

Differentiate sine x using the limit definition and unit circle geometry to show the derivative of sine is cosine, using arc length and sandwich theorem for limiting behavior.

Derive inverse functions using the inverse derivative rule, linking the derivatives of f and its inverse as reciprocals. Explore intuitive reasoning with rate examples and apply to the nth root.

Apply the inverse derivative rule to differentiate the nth root function. Treat x as y^n, substitute, and obtain f'(x) = (1/n) x to the one over n minus one.

Apply the inverse rule to differentiate the log function and derive that log x is 1/x when e^y equals x; note arcsine as the inverse of sine.

Explore arcsine by differentiating its inverse relationship with x using the inverse rule, obtaining the derivative 1/sqrt(1 - x^2) with sign depending on the angle range.

Explore the challenges of direct integration and how initial conditions affect constants. Connect lower and upper bounds, indefinite integrals, and velocity to the net change in y as x changes.

Explore the fundamental theorem of calculus, linking differentiation and integration through a two-part circle, recovering antiderivatives with constants and equating definite integrals to end-value differences.

Explore how integration reduces to differentiation by identifying antiderivatives, using the fundamental theorem of calculus. Learn with exponential, polynomial, nth root, and sine and cosine examples to find integrals.

Learn how basic operations—add, subtract, multiply, and divide—combine functions into complex forms, and differentiate them using the product rule, quotient rule, and the chain rule for composition.

Apply the chain rule by viewing a function as two layers with an intermediate u, relating y to x through h and g via the product of their derivatives.

Explore integration by parts and substitution, linking them to differentiation rules. Identify inner and outer layers in composite functions to apply the reverse chain rule.

Finish the integration table using integration by parts on arcsine and the inverse chain rule with u = 1 − x^2, and apply parts to log x.