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

For example,

## 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

For example,

The power function rule is extremely powerful! You can use it with a variety of exponents. For example,

can be rewritten as

Be careful with this last derivative. When a variable with an exponent appears in the denominator, such as x^{3} in the previous equation, the variable can be moved to the numerator, but the exponent becomes negative. So, 4/x^{3} becomes 4x^{-3}. Then when you take the derivative, make sure you subtract 1 from –3 to get –4.

As another example, consider

can be written as

You may remember that square roots are fractional exponents, or the 0.5 (one-half) power.

Finally, note that

## 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,

For example,

Then the derivatives of *TR* and *TC* with respect to *q* are

Using the sum-difference rule

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

For example, if

In this equation, *u = x** ^{3}* and

*v = (9 + 4x - 7x*

^{2}*).*Thus, the derivative of

*u*with respect to

*x*is

And the derivative of *v* with respect to *x* is

Then

## 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,

For example, if the original quotient is

In this quotient, *u* = *x*^{3} and *v* = (5*x* - 2). The derivative of *u* with respect *x* is

And the derivative of *v* with respect to *x* is

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

The denominator is *v*^{2} or

Substituting everything into the quotient rule yields

## 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

For example, if

Then

Substituting *y* = (3*x*^{2} – 5*x* +7) into *dz/dx* yields

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*.