1. 3.1 – Rules for the Derivative
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)
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

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
f(x)=2x and f(x)=3x
f(x) = ax , then f’(x) = f’(0) f(x)