The following graph is a trigonometric function that represents the periodic temperatures of Chicago from 2011-2012.
The trigonometric equations above represent Chicago temperatures as a function of time. Find the derivative equation of both functions.
STEP ONE: DEFINE Rule NEEDED
The following equations contain a function within a function. The outside function being 28.525cos(x)+49.775, and the inside function being (pi/6)x-(7pi/6) . (The second equation having the outside function of 28.525sin(x)+49.775 and inside function of (pi/6)x-(2pi/6)). In order to find the derivative function the chain rule must be applied.
Step two: find derivatives of inside/outside functions
In order to find the derivative of the outside function we must know the trigonometric property of derivatives. For the following 2 equations there are 2 trigonometric properties that must be known. They are the following: d/dx COS(x)= -SIN(x) AND
d/dx SIN(x)= COS(x). Knowing this we can then find the derivatives of our outside functions.
d/dx SIN(x)= COS(x). Knowing this we can then find the derivatives of our outside functions.
Derivative of Inside Functions
The following inside functions are linear. Therefore the derivative of the function is simply the slope of the line. (The constant that is being multiplied by x is the slope of a linear line.) In this case both functions will have a derivative of pi/6.
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.
Example Problem with derivative equations
Find the rate at which the temperature changes in the 5th month of Chicago 2011.