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

h

afhafaf

h

)()(lim)('

0

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)– 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

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

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