324
Chapter 3 | Derivatives
Solution Use the derivative of the natural exponential function, the quotient rule, and the chain rule.
⎛ ⎝ e x
⎞ ⎠ x · x −1· e x 2 x 2
2
·2
y ′ =
Apply the quotient rule.
2 ⎛
⎝ 2 x 2 −1 ⎞ ⎠ x 2
e x
=
Simplify.
Find the derivative of h ( x ) = xe 2 x .
3.50
Example 3.76 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. Solution 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.
3.51 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?
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.
Theorem 3.15: The Derivative of the Natural Logarithmic Function If x >0 and y = ln x , then
(3.30) dy dx = 1 x . More generally, let g ( x ) be a differentiable function. For all values of x for which g ′( x ) >0, the derivative of
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