Introduction to Calculus 1: Differentiation

Review algebra basics by factoring polynomials, extracting the greatest common factor, and applying differences of squares and cubes, then factor quadratics to rewrite them in factored form.

Learn how functions map inputs to outputs and distinguish one-to-one from many-to-one mappings. Graph linear, quadratic, cubic, rational, and square root functions, and use slope and y-intercept to describe lines.

Explore domain and range concepts, unite and intersect sets, and apply examples like rational functions and square roots to identify input and output values.

Identify horizontal and vertical asymptotes in functions like 1/x and e^x, noting x=0 yields a vertical asymptote and y=0 a horizontal asymptote. Preview methods to find them, including composition functions.

Learn how composition functions create outputs by feeding one function into another. Use examples like f(x)=sqrt(x) and g(x)=2x+1 to form the composition f(g(x)).

Explore how inverse functions swap inputs and outputs to map from the range to the domain, using F^-1 notation. See that only one-to-one functions have inverses, while X^2 has none.

Explore limits and continuity, including left and right limits, vertical and horizontal asymptotes, and pivotal algebraic techniques. Define continuity and distinguish removable versus jump discontinuities using one-sided limits.

Learn how secant lines measure average rate of change between close points and how the tangent line is the limit of secant slopes as points approach.

Explore the derivative function as the slope of tangents, defined by a limit, with examples like f(x)=x^2 and g(x)=square root of x, and learn to derive and tangent-line equations.

Develop and apply the power rule to find derivatives of x^n, including constants, and recognize patterns from examples like x^3, sqrt(x), and 1/x^8, with constants' derivatives equal to zero.

Explore derivative notation across forms such as f'(x), y', dy/dx, and df/dx, and apply the power rule f'(x) = p x^(p-1) for functions, with constant derivatives equal to zero.

Learn the constant multiple rule, pulling constants outside derivatives, and apply the power rule to negative and fractional exponents, with a quick limit-based justification.

Learn the sum and difference rules for derivatives: the derivative of a sum equals the sum of the derivatives, with constant multiple and power rules shown via 3x^2 + 4x.

Apply the product rule to derivatives of products, confirming (f g)' = f' g + f g', with examples like x^3 times x^2, and prove it via the limit definition.

Apply quotient rule to differentiate a ratio F(x)/G(x) by computing (F'(x)G(x) - F(x)G'(x)) / [G(x)]^2, as shown with examples like X^7/X^4 and Y=(X+1)/(X-1).

Explore differentiability by analyzing limits and tangent behavior to determine whether the derivative exists; learn that differentiability implies continuity, but continuity does not guarantee differentiability, citing the absolute function.

The normal line is perpendicular to the tangent line. With f(x)=4x^4+x+1 at x=1, the tangent slope is 17, so the normal line is y=-1/17 x+103/17.

Explore higher order derivatives from the second to the sixth for a polynomial, noting how successive differentiation yields zeros, with Leibniz notation and a preview of composition functions.

Explore the chain rule for differentiating compositions by pairing the outer derivative with the inner input, using inside functions, and applying power, product, and quotient rules.

Learn how implicit differentiation derives dy/dx for equations implicitly defining y, such as x^2 + y^2 = 25 and y^2 = 2x, by differentiating both sides and solving for y'.

Use implicit differentiation on x^3 + y^3 = x + 1 to find dy/dx. Substitute (0,1) to get the tangent line y = (1/3)x + 1.

Master derivatives of transcendental functions by reviewing trigonometric basics, including radians conversion, unit circle coordinates, and core identities such as sin^2+cos^2=1 and 1+tan^2=sec^2.

Derive trigonometric derivatives, including sine, cosine, tangent, secant, and cosecant, using product, quotient, and chain rules, and note the two special elements with implicit differentiation examples.

Explains derivatives of trig and composite functions using the chain rule, including sin(cos x) and (sec x + 5)^(1/3); demonstrates implicit differentiation to solve for y'.

Learn how inverse trig functions exist only on restricted domains to yield one-to-one mappings. Master arcsin, arccos, and arctan with sine, cosine, and tangent examples.

Derive the derivatives of inverse trig functions—arcsin, arccos, and arctan—using implicit differentiation, right-triangle reasoning, and Pythagorean theorem. Apply product and chain rules to composite expressions like x^3 arcsin(x) and arctan(2x^2).

Review exponential and log rules, including base, power, product, quotient, and change-of-base formulas, and relate them to exponential and logarithmic graphs as inverses.

Derive the derivatives of exponential and logarithmic functions, apply product and chain rules, and use the change of base to differentiate log base b of x.

Master log differentiation for functions with variable bases and exponents, derive Y' via implicit differentiation, and recognize e^x as a special case.

Explore log differentiation for functions where both base and exponent depend on x, using implicit differentiation. Work through examples like y=(2x−e^{6x})^{cos(5x)}, y=(1/x)^{ln x}, and y=x^{1/x}.

Master the fundamentals of differentiation, from the derivative definition and constants to product, quotient, and chain rules, plus implicit differentiation, logarithmic and exponential derivatives, and basic trig rules.

Explore real-life uses of derivatives in part three of calculus one: applications of derivatives. Two bonus lectures from part three and a level review support future students.

Explore limits and indeterminate forms, apply l'Hôpital's rule to 0/0 and infinity/infinity cases, and compare algebraic simplifications like factoring and removing radicals as alternatives.

Explore the meaning of derivatives as rates of change, including average and instantaneous rates, with the example f(x)=x^2 and applications to velocity and acceleration.