![]() SolutionĬonveniently, this is already factored. Rational functions then are functions written as fractions of polynomial functions in the form \(f(x) = \frac\). In mathematics, rational means "ratio" or can be written as a fraction. Rational functions are like the one above in the introduction. This lesson is about rational functions which have variables in the denominator. The last few lessons have been about polynomial functions which have non-negative integers for exponents. Written without a variable in the denominator, this function will contain a negative integer power. Many other applications require finding averages in a similar way. The average cost function for this situation is The average cost for producing x items is found by dividing the cost function by the number of items, x. This indicates that each item costs $125 and there is a $2000 initial cost to setup the production floor. In a particular factory, the cost is given by the equation C( x) = 125 x + 2000. In factories, the cost of making a product is dependent on the number of items, x, produced. Find the domains of rational functions.įigure 1: Factory floor (credit pixabay/Tama66).We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative.The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.The derivative of the difference of a function \(f\) and a function \(g\) is the same as the difference of the derivative of \(f\) and the derivative of \(g\).The derivative of the sum of a function \(f\) and a function \(g\) is the same as the sum of the derivative of \(f\) and the derivative of \(g\).The derivative of a constant \(c\) multiplied by a function \(f\) is the same as the constant multiplied by the derivative. ![]() The derivative of a power function is a function in which the power on \(x\) becomes the coefficient of the term and the power on \(x\) in the derivative decreases by 1.The derivative of a constant function is zero. ![]()
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