Find values of derivatives using limits

Learning Objectives Define the derivative as the limit of the difference quotient. Calculate the derivative of a function at a specific point using the limit definition. Find the general derivative function, f'(x), for polynomial functions using the limit definition. Find the derivative function, f'(x), for simple radical and rational functions using the limit definition. Connect the value of the derivative at a point to the slope of the tangent line at that point. Use the alternative form of the derivative definition to find the derivative at a specific point. Ever wonder how a police officer's radar gun calculates your car's exact speed at one instant, not just your average speed over a trip? 🚓 That's the power of instantaneous rates of change! I...

TermDefinitionExample Secant LineA line that passes through two distinct points on a curve. The slope of the secant line represents the average rate of change of the function between those two points.For the curve y = x², the line passing through the points (1, 1) and (3, 9) is a secant line. Its slope is (9-1)/(3-1) = 4. Tangent LineA line that touches a curve at a single point and has the same direction as the curve at that point. The slope of the tangent line represents the instantaneous rate of change of the function at that point.For the curve y = x², the tangent line at the point (1, 1) has a slope of 2. Difference QuotientThe expression (f(x + h) - f(x)) / h. It represents the slope of the secant line between the points (x, f(x)) and (x + h, f(x + h)).For f(x) = x², the difference...

The Limit Definition of the Derivative f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} This is the primary formula used to find the derivative as a function, f'(x). It calculates the instantaneous rate of change for any 'x' by finding the limit of the slope of the secant line as the distance between the points, 'h', shrinks to zero. The Alternative Form of the Derivative at a Point f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} This is an alternative formula used to find the derivative at a single, specific point, x = a. It is conceptually the same as the primary definition but can sometimes be algebraically simpler for certain functions.