Mastering Derivatives: A Complete Calculus Course

The derivative is the slope of the tangent or instantaneous rate of change, defined as the limit of (f(x+h)-f(x))/h as h approaches zero, with examples for x^2 and sqrt(T).

Apply derivative rules for polynomial functions: constant, power, constant multiple, and sum and difference rules, with examples like x^4, sqrt(x), and linear terms to compute derivatives.

Master the product rule for differentiating the product of two functions, using F'G + FG' with examples. Explore the power of a function and differentiating a function inside another.

Derive the quotient rule by rewriting a ratio as a product with a negative power, then apply the product rule. Use the rule on derivatives and a tangent line example.

Apply the chain rule to differentiate composite functions, using h(x)=F(G(x)) with h′(x)=F′(G(x))·G′(x). Explain with examples f(x)=x^2 and g(x)=x+4, showing 2(x+4), and y=(x^2−5)^7 yielding 14x(x^2−5)^6, noting F∘G ≠ G∘F.

Explore higher order derivatives by differentiating a cubic position function to obtain velocity and acceleration, and apply these concepts to analyze motion on a straight line with time-dependent position functions.

Find the maximum or minimum of a function on a restricted interval by deriving the function, solving for turning points, and evaluating endpoints and turning points within the interval.

Apply optimization to real-world problems by formulating area and volume functions from fencing and metal sheet problems, using derivatives to find maxima under constraints.

Learn to find intervals of increase and decrease by solving the derivative, using tangent slopes to classify increasing or decreasing behavior, and identifying critical points.

Use the first derivative test to find local maxima and minima by locating critical points where the derivative is zero or undefined, then assess increasing and decreasing intervals.

Identify vertical, horizontal, and oblique asymptotes in rational functions, finding verticals by denominator zero, horizontals by degrees, and obliques via long division.

Use the second derivative to analyze concavity and identify points of inflection, testing where concavity changes sign. Apply the second derivative test to locate local maxima and minima.

Learn a step-by-step curve sketching method, combining x/y intercepts, vertical and horizontal asymptotes, and first/second derivative tests to locate critical points and concavity, illustrated with f(x)=(x^2+4)/(x^2-4).

Explore curve sketching techniques by analyzing intercepts, vertical and horizontal asymptotes, and function behavior across intervals. Learn to determine increasing/decreasing and concavity using an interval table.

Master the derivative of exponential functions, including bases B and e, and identify increasing or decreasing behavior. Apply the chain rule to composite exponents with practical examples.

Explore derivatives of sinusoidal functions like sin x and cos x, applying chain and product rules, then derive the tangent to y = 2x cos 2x at pi/2.

Derive tan x with the quotient rule, confirming its derivative is sec^2 x, and apply chain, product, and sine-cosine differentiation concepts.