Math 1304 Calculus I

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

2. 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

3. 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.

4. 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?

5. 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) • Proof? • Constant multiple: If f(x) = c g(x), then f’(x) = c g’(x) • Proof? • More? – coming soon • Sums of several functions • Linear combinations • Product • Quotient • Composition

6. 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

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

8. Exponentials • Exponentials: If f(x) = ax, then f’(x) = (ln a) ax Discussion If f(x) = ax , then f’(x) = f’(0) f(x) Proof? Special cases f(x)=2x and f(x)=3x and f(x)=ex