Here is a brief refresher for some of the important rules of calculus differentiation for managerial economics. While calculus is not necessary, it does make things easier.
Constant function rule
If variable y is equal to some constant a, its derivative with respect to x is 0, or if
![image0.jpg](https://www.dummies.com/wp-content/uploads/373185.image0.jpg)
For example,
![image1.jpg](https://www.dummies.com/wp-content/uploads/373186.image1.jpg)
Power function rule
A power function indicates that the variable x is raised to a constant power k.
The derivative of y with respect to x equals k multiplied by x raised to the k-1 power, or
![image2.jpg](https://www.dummies.com/wp-content/uploads/373187.image2.jpg)
For example,
![image3.jpg](https://www.dummies.com/wp-content/uploads/373188.image3.jpg)
The power function rule is extremely powerful! You can use it with a variety of exponents. For example,
![image4.jpg](https://www.dummies.com/wp-content/uploads/373189.image4.jpg)
can be rewritten as
![image5.jpg](https://www.dummies.com/wp-content/uploads/373190.image5.jpg)
Be careful with this last derivative. When a variable with an exponent appears in the denominator, such as x3 in the previous equation, the variable can be moved to the numerator, but the exponent becomes negative. So, 4/x3 becomes 4x-3. Then when you take the derivative, make sure you subtract 1 from –3 to get –4.
As another example, consider
![image6.jpg](https://www.dummies.com/wp-content/uploads/373191.image6.jpg)
can be written as
![image7.jpg](https://www.dummies.com/wp-content/uploads/373192.image7.jpg)
You may remember that square roots are fractional exponents, or the 0.5 (one-half) power.
Finally, note that
![image8.jpg](https://www.dummies.com/wp-content/uploads/373193.image8.jpg)
Sum-difference rule
Assume there are two functions, TR = g(q) and TC = h(q).
You may think of the variable TR as total revenue, the variable TC as total cost, and the variable q as the quantity of the product produced. The symbol g in the total revenue function and the symbol h in the total cost function mean that the relationship between q and total revenue is different from the relationship between q and total cost.
Further, assume that the variable ð (profit) is a function of both TR and TC, so
ð = <i>TR</i> – <i>TC</i>.
The derivative of ð with respect to q equals the sum (the functions can be added or subtracted) of the derivatives of TR and TC with respect to q, or,
![image9.jpg](https://www.dummies.com/wp-content/uploads/373194.image9.jpg)
For example,
![image10.jpg](https://www.dummies.com/wp-content/uploads/373195.image10.jpg)
Then the derivatives of TR and TC with respect to q are
![image11.jpg](https://www.dummies.com/wp-content/uploads/373196.image11.jpg)
Using the sum-difference rule
![image12.jpg](https://www.dummies.com/wp-content/uploads/373197.image12.jpg)
Although in the example the two functions were subtracted, remember that the sum difference rule also works when functions are added.
Product rule
Assume you have two functions, u = g(x) and v = h(x). Further, assume that y = u × v.
The derivative of y with respect to x equals the sum of u multiplied by the derivative of v and v multiplied by the derivative of u, or if
![image13.jpg](https://www.dummies.com/wp-content/uploads/373198.image13.jpg)
For example, if
![image14.jpg](https://www.dummies.com/wp-content/uploads/373199.image14.jpg)
In this equation, u = x3 and v = (9 + 4x - 7x2). Thus, the derivative of u with respect to x is
![image15.jpg](https://www.dummies.com/wp-content/uploads/373200.image15.jpg)
And the derivative of v with respect to x is
![image16.jpg](https://www.dummies.com/wp-content/uploads/373201.image16.jpg)
Then
![image17.jpg](https://www.dummies.com/wp-content/uploads/373202.image17.jpg)
Quotient rule
A quotient refers to the result obtained when one quantity, in the numerator, is divided by another quantity, in the denominator.
Assume you have two functions, u = g(x) and v = h(x). So, u is the quantity in the numerator, and it’s a function g of x. And v is the quantity in the denominator, and it’s a different function of x as represented by h. In addition, assume that y = u/v. So y is the quotient of u divided by v.
The derivative of y with respect to x has two components in its numerator. The first component is the original equation for v multiplied by the derivative of u taken with respect to x, du/dx. From that amount, you subtract the numerator’s second component, the original equation u multiplied by the derivative of v taken with respect to x, dv/dx.
The dominator of this derivative is simply the original equation, v, squared. Thus,
![image18.jpg](https://www.dummies.com/wp-content/uploads/373203.image18.jpg)
For example, if the original quotient is
![image19.jpg](https://www.dummies.com/wp-content/uploads/373204.image19.jpg)
In this quotient, u = x3 and v = (5x - 2). The derivative of u with respect x is
![image20.jpg](https://www.dummies.com/wp-content/uploads/373205.image20.jpg)
And the derivative of v with respect to x is
![image21.jpg](https://www.dummies.com/wp-content/uploads/373206.image21.jpg)
Thus, the first component of the numerator is v multiplied du/dx. From that, you subtract the second component of the numerator, which is u multiplied by dv/dx, or
![image22.jpg](https://www.dummies.com/wp-content/uploads/373207.image22.jpg)
The denominator is v2 or
![image23.jpg](https://www.dummies.com/wp-content/uploads/373208.image23.jpg)
Substituting everything into the quotient rule yields
![image24.jpg](https://www.dummies.com/wp-content/uploads/373209.image24.jpg)
Chain rule
You’re almost there, and you’re probably thinking, “Not a moment too soon.” Just one more rule is typically used in managerial economics — the chain rule.
For the chain rule, you assume that a variable z is a function of y; that is, z = f(y). In addition, assume that y is a function of x; that is, y = g(x). The derivative of z with respect to x equals the derivative of z with respect to y multiplied by the derivative of y with respect to x, or
![image25.jpg](https://www.dummies.com/wp-content/uploads/373210.image25.jpg)
For example, if
![image26.jpg](https://www.dummies.com/wp-content/uploads/373211.image26.jpg)
Then
![image27.jpg](https://www.dummies.com/wp-content/uploads/373212.image27.jpg)
Substituting y = (3x2 – 5x +7) into dz/dx yields
![image28.jpg](https://www.dummies.com/wp-content/uploads/373213.image28.jpg)
With this last substitution, you remove the third variable y from the derivative, and as a result, you have a function for dz/dx only in terms of x.