Derivative measures rate of change of function's output relative to input. Derivative is slope of tangent line at point of interest. Derivative definition derived from slope formula for lines
Derivative of ln(x) is defined as limit of ln(x+h) - ln(x)h. Simplifying gives limit of (1+hx)1h = e1x. Final result is limh→0hx/h = 1x
Tangent is a straight line that touches a curve at a given point. Tangent line passes through point (c, f(c)) with slope f'(c). Tangent line is best straight-line approximation to curve at point of tangency. Tangent plane touches surface at given point
Limit definition as function value approach is too subjective. Example shows limit at x=2 should be 0 due to jump discontinuity. Example shows limit at x=0 should be 0 despite damped amplitude
Function crosses X-axis at f(0) = 0. Y-axis intersection occurs at x = 0
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