3.1 – Rules for the Derivative Math 1304 Calculus I 3.1 – Rules for the Derivative

Definition of Derivative The definition from the last chapter of the derivative of a function is: Definition: The derivative of a function f at a number a, denoted by f’(a) is given by the formula

A Faster Systematic Way Use rules Use formulas for basic functions such as constants, power, exponential, and trigonometric. Use rules for combinations of these functions.

Derivatives of basic functions Constants: If f(x) = c, then f’(x) = 0 Proof? Powers: If f(x) = xn, then f’(x) = nxn-1 Discussion?

Rules for Combinations Sum: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) Proof? Difference: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) Constant multiple: If f(x) = c g(x), then f’(x) = c g’(x) More? – coming soon Sums of several functions Linear combinations Product Scalar multiple Quotient Composition

Start of a good working set of rules Constants: If f(x) = c, then f’(x) = 0 Powers: If f(x) = xn, then f’(x) = nxn-1 Exponentials: If f(x) = ax, then f’(x) = (ln a) ax Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x) Sum: If f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) Difference: If f(x) = g(x) - h(x), then f’(x) = g’(x) - h’(x) Multiple sums: the derivative of sum is the sum of derivatives (derivatives apply to polynomials term by term) Linear combinations: derivative of linear combination is linear combination of derivatives Monomials: If f(x) = c xn, then f’(x) = n c xn-1 Polynomials: term by term monomials

Examples f(x)= 2x3 +3x2 + 5x + 1, find f’(x) Find d/dx (x5 + 3 x4 – 5x3 + x2 + 4) y = 3x2 + 20, find y’

Exponentials Exponentials: If f(x) = ax, then f’(x) = (ln a) ax Discussion f(x)=2x and f(x)=3x f(x) = ax , then f’(x) = f’(0) f(x)