The derivative of ln(x) is 1/x. Natural log x is logarithmic function with base Euler's Number. Derivative shows change in function value with respect to input variable
Local maximum occurs when function's y-coordinate is larger than nearby points. Local minimum occurs when function's y-coordinate is smaller than nearby points. Local extremum is either maximum or minimum. Tangent lines must be horizontal at local extrema
Limit of (1-cos(x))/x² approaches 0 as x approaches 0. Direct evaluation yields indeterminate result 0/0
Linear approximation approximates function values using tangent lines. Tangent line slope equals derivative f'(a) at given point. Formula: L(x) = f(a) + f'(a) (x - a)
A cubic function is a polynomial of degree 3 in the form f(x) = ax³ + bx² + cx + d. It can have 1 or 3 real roots, with 0 or 2 complex roots. The basic cubic function is f(x) = x³
Rates of change can be calculated using the formula Df/Dt = (f(x) - f(x))/x. Tangents to curves are found using the formula Dy/Dt = (f(x) - f(x))/x. Slope of a curve is calculated using Ds/Dt = (f(x) - f(x))/x