Section 3.1 Derivatives of Polynomials and Exponential Functions  Goals Learn formulas for the derivatives ofLearn formulas for the derivatives of  Constant functions  Power functions  Exponential functions Learn to find new derivatives from old:Learn to find new derivatives from old:  Constant multiples  Sums and differences

Constant Functions  The graph of the constant function f(x) = c is the horizontal line y = c … which has slope 0,which has slope 0, so we must have f (x) = 0 (see the next slide).so we must have f (x) = 0 (see the next slide).  A formal proof is easy:

Constant Functions (cont’d)

Power Functions  Next we look at the functions f(x) = x n, where n is a positive integer.  If n = 1, then the graph of f(x) = x is the line y = x, which has slope 1, so f (x) = 1.  We have already seen the cases n = 2 and n = 3 :

Power Functions (cont’d)  For n = 4 we find the derivative of f(x) = x 4 as follows:

Power Functions (cont’d)  There seems to be a pattern emerging!  It appears that in general, if f(x) = x n, then f (x) = nx n - 1.  This turns out to be the case:

Power Functions (cont’d)  We illustrate the Power Rule using a variety of notations:  It turns out that the Power Rule is valid for any real number n, not just positive integers:

Power Functions (cont’d)

Constant Multiples  The following formula says that the derivative of a constant times a function is the constant times the derivative of the function:

Sums and Differences  These next rules say that the derivative of a sum (difference) of functions is the sum (difference) of the derivatives:

Example

Exponential Functions  If we try to use the definition of derivative to find the derivative of f(x) = a x, we get:  The factor a x doesn’t depend on x, so we can take it in front of the limit:

Exponential (cont’d)  Notice that the limit is the value of the derivative of f at 0, that is,

Exponential (cont’d)  This shows that… if the exponential function f(x) = a x is differentiable at 0,if the exponential function f(x) = a x is differentiable at 0, then it is differentiable everywhere andthen it is differentiable everywhere and f (x) = f (0)a x  Thus, the rate of change of any exponential function is proportional to the function itself.

Exponential (cont’d)  The table shown gives numerical evidence for the existence of f (0) when a = 2 ; here apparentlya = 2 ; here apparently f (0) ≈ 0.69 a = 3 ; here apparentlya = 3 ; here apparently f (0) ≈ 1.10

Exponential (cont’d)  So there should be a number a between 2 and 3 for which f (0) = 1, that is,  But the number e introduced in Section 1.5 was chosen to have just this property!  This leads to the following definition:

Exponential (cont’d)  Geometrically, this means that of all the exponential functions y = a x,of all the exponential functions y = a x, the function f(x) = e x is the one whose tangent at (0, 1) has a slope f (0) that is exactly 1.the function f(x) = e x is the one whose tangent at (0, 1) has a slope f (0) that is exactly 1. This is shown on the next slide:This is shown on the next slide:

Exponential (cont’d)

 This leads to the following differentiation formula:  Thus, the exponential function f(x) = e x is its own derivative.

Example  If f(x) = e x – x, find f (x) and f  (0).  Solution The Difference Rule gives  Therefore

Solution (cont’d)  Note that e x is positive for all x, so f  (x) > 0 for all x.  Thus, the graph of f is concave up. This is confirmed by the graph shown.This is confirmed by the graph shown.

Review  Derivative formulas for polynomial and exponential functions  Sum and Difference Rules  The natural exponential function e x