# Meaning of Derivatives

Applying the definition of a derivative and understanding different notations. Creating the graph of a function and its derivative. Finding the right‐hand and left‐hand derivative of a function.Connecting derivative of a function with tangent line.

Determine average rates of change using difference quotients. The difference quotients [f(a + h) – f(a)] / h and [f(x) – f(a)] / (x – a) express the average rate of change of a function over an interval. The instantaneous rate of change of a function at x = a can be expressed by [f(a + h) – f(a)] / h for lim h → 0 or [f(x) – f(a)] / (x – a) for lim x → a, provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted f′(a ). The derivative of f is the function whose value at x is [f(x + h) – f(x)] / h for lim h → 0, provided this limit exists. For y = f(x), notations for the derivative include dy/dx, f′(x), and y′. The derivative can be represented graphically, numerically, analytically, and verbally. Determine the equation of a line tangent to a curve at a given point. The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point. Estimate derivatives. The derivative at a point can be estimated from information given in tables or graphs. Technology can be used to calculate or estimate the value of a derivative of a function at a point.