Find the rate at which the population growth changes in the years of 1920 and 1975.
Step One: DefinE Rule needed
In order to find the derivative of an natural exponential functions the chain rule must also be applied. Also it must be known that the derivative of f(x) = e^x is f '(x) = e^x. Now that we covered the rules that must be known, we can then apply the chain rule to find the derivative function of each equation.
Step two: Find derivative of outside/inside functions.
Since we have defined that the derivative of e^x is simple e^x, we can then find the derivatives of our outside functions.
Derivative of Inside Functions
The Inside functions are linear so the derivative of these functions are simply the slope. (The constant that is being multiplied by x is the slope of a linear line.)
STEP three: PLUG IT IN
Now that we have the derivatives of the outside and inside functions, we can then apply the chain rule to find a derivative function for each equation.
Finding Rate of Change in 1920
The year 1920 falls in-between 1900-1950. We therefore will need to plug in the T years away from 1900 to 1920 in order to find the rate at which the population growth of Chicago changes in 1920.
FINDING RATE OF CHANGE IN 1975
The year 1975 falls in-between 1950-1990. We therefore will need to plug in the T years away from 1950 to 1975 in order to find the rate at which the population growth of Chicago changes in 1975.