<< Chapter < Page Chapter >> Page >
b −0.00001 1 −0.00001 < B ( 0 ) < b 0.00001 1 0.00001 .

See the following table.

Estimating a value of e
b b −0.00001 1 −0.00001 < B ( 0 ) < b 0.00001 1 0.00001 b b −0.00001 1 −0.00001 < B ( 0 ) < b 0.00001 1 0.00001
2 0.693145 < B ( 0 ) < 0.69315 2.7183 1.000002 < B ( 0 ) < 1.000012
2.7 0.993247 < B ( 0 ) < 0.993257 2.719 1.000259 < B ( 0 ) < 1.000269
2.71 0.996944 < B ( 0 ) < 0.996954 2.72 1.000627 < B ( 0 ) < 1.000637
2.718 0.999891 < B ( 0 ) < 0.999901 2.8 1.029614 < B ( 0 ) < 1.029625
2.7182 0.999965 < B ( 0 ) < 0.999975 3 1.098606 < B ( 0 ) < 1.098618

The evidence from the table suggests that 2.7182 < e < 2.7183 .

The graph of E ( x ) = e x together with the line y = x + 1 are shown in [link] . This line is tangent to the graph of E ( x ) = e x at x = 0 .

Graph of the function ex along with its tangent at (0, 1), x + 1.
The tangent line to E ( x ) = e x at x = 0 has slope 1.

Now that we have laid out our basic assumptions, we begin our investigation by exploring the derivative of B ( x ) = b x , b > 0 . Recall that we have assumed that B ( 0 ) exists. By applying the limit definition to the derivative we conclude that

B ( 0 ) = lim h 0 b 0 + h b 0 h = lim h 0 b h 1 h .

Turning to B ( x ) , we obtain the following.

B ( x ) = lim h 0 b x + h b x h Apply the limit definition of the derivative. = lim h 0 b x b h b x h Note that b x + h = b x b h . = lim h 0 b x ( b h 1 ) h Factor out b x . = b x lim h 0 b h 1 h Apply a property of limits. = b x B ( 0 ) Use B ( 0 ) = lim h 0 b 0 + h b 0 h = lim h 0 b h 1 h .

We see that on the basis of the assumption that B ( x ) = b x is differentiable at 0 , B ( x ) is not only differentiable everywhere, but its derivative is

B ( x ) = b x B ( 0 ) .

For E ( x ) = e x , E ( 0 ) = 1 . Thus, we have E ( x ) = e x . (The value of B ( 0 ) for an arbitrary function of the form B ( x ) = b x , b > 0 , will be derived later.)

Derivative of the natural exponential function

Let E ( x ) = e x be the natural exponential function. Then

E ( x ) = e x .

In general,

d d x ( e g ( x ) ) = e g ( x ) g ( x ) .

Derivative of an exponential function

Find the derivative of f ( x ) = e tan ( 2 x ) .

Using the derivative formula and the chain rule,

f ( x ) = e tan ( 2 x ) d d x ( tan ( 2 x ) ) = e tan ( 2 x ) sec 2 ( 2 x ) · 2 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Combining differentiation rules

Find the derivative of y = e x 2 x .

Use the derivative of the natural exponential function, the quotient rule, and the chain rule.

y = ( e x 2 · 2 ) x · x 1 · e x 2 x 2 Apply the quotient rule. = e x 2 ( 2 x 2 1 ) x 2 Simplify.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the derivative of h ( x ) = x e 2 x .

h ( x ) = e 2 x + 2 x e 2 x

Got questions? Get instant answers now!

Applying the natural exponential function

A colony of mosquitoes has an initial population of 1000. After t days, the population is given by A ( t ) = 1000 e 0.3 t . Show that the ratio of the rate of change of the population, A ( t ) , to the population, A ( t ) is constant.

First find A ( t ) . By using the chain rule, we have A ( t ) = 300 e 0.3 t . Thus, the ratio of the rate of change of the population to the population is given by

A ( t ) = 300 e 0.3 t 1000 e 0.3 t = 0.3 .

The ratio of the rate of change of the population to the population is the constant 0.3.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

If A ( t ) = 1000 e 0.3 t describes the mosquito population after t days, as in the preceding example, what is the rate of change of A ( t ) after 4 days?

996

Got questions? Get instant answers now!

Derivative of the logarithmic function

Now that we have the derivative of the natural exponential function, we can use implicit differentiation to find the derivative of its inverse, the natural logarithmic function.

The derivative of the natural logarithmic function

If x > 0 and y = ln x , then

d y d x = 1 x .

More generally, let g ( x ) be a differentiable function. For all values of x for which g ( x ) > 0 , the derivative of h ( x ) = ln ( g ( x ) ) is given by

h ( x ) = 1 g ( x ) g ( x ) .
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask