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) = x n, then f’(x) = nx n-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) = x n, then f’(x) = nx n-1 Exponentials: If f(x) = a x, then f’(x) = (ln a) a x 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 x n, then f’(x) = n c x n-1 Polynomials: term by term monomials

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

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