Math Help

Derivatives – Using limits and known derivatives of functions

In a Nut Shell: The notion of a “derivative” of a function is fundamental to differential

calculus. Derivatives relate the “change” of the dependent variable, f(x), over a “small”

change in the independent variable, say x. The derivative of f with respect to x typically

appears as df/dx. The definition of the derivative uses limits in its calculation.

Definition: The derivative of a function, f(x), with respect to the independent variable x is:

df/dx = f ’(x) = lim [ f(x + h) - f(x)]/ h

h → 0

Some common applications of derivatives are:

Slope of tangent line to curve = m = f ’(x) = dy/dx

Rate of change with time, t, of a quantity, Q(t) is Q ’(t) = dQ/dt

Speed of a falling object = v(t), y(t) = position, v(t) = dy/dt

The Power and Product rules for derivatives of a function of x are:

Power rule for positive integer n: f(x) = xⁿ, df/dx = f ’(x) = n x^n-1

Power rule for: f(x) = x ^-n, df/dx = f ’(x) = ˗ n x ^-n-1

Product rule for two functions f and g: d/dx[f(x)g(x)] = f ’(x) g(x) + f(x) g ’(x)

You will be calculating the derivatives of products, quotients, and various types of

simple functions.