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Properties of Derivatives

**Linearity · Linear Combination Rule · Example using Linear Combination Rule ·
The Product Rule · Definition
of Product Rule · Example using Product Rule
· The Quotient Rule · Definition of the Quotient Rule · Example using Quotient Rule · Definition of the
Chain Rule (function of a function version) · Definition of the Chain Rule
(composition version) · Example of the Chain Rule · ****Recommended Books**

This article covers laws that allow us to build up derivatives of complicated functions from simpler ones. These laws form part of the everyday tools of differential calculus.

Suppose f(x) and g(x) are differentiable functions and a and b are real numbers. Then the function

is differentiable and the derivative is

This **linearity law** applies to many, many
mathematical processes, so much so that we feel it is important to emphasize its nature
with alternate notation.

Let us represent differentiation with respect to
x by applying an **operator** D_{x} to a function:

The last notation on the right is the older, traditional notation for derivative and is included for completeness. Using this notation, the linearity law becomes

Linearity is so common that it behooves the reader to thoroughly digest this concept.

Example using Linear Combination Rule

Find the derivative of

Solution

Using linearity,

We can extend the notion of linearity to cover any number of constants and functions.

Let

Then

This rule can be applied for any finite number of terms.

This rule is used to find the derivative of a product of two functions.

Let

Then

In short,

Use the product rule to find the derivative of

Solution

Here,

Thus

Note: Although this can be simplified, most professors (and graders!) prefer that you leave it as it is – that way, use of the product rule is clear. Also, astute readers probably observed that we could have more easily simplified h(x) algebraically first and then applied the linear combination rule. While that may be true for this example, in later sections, we will cover functions where the only practical way to do the derivative is by use of the product rule. In some cases, the easiest way to do a derivative is not the best way.

The quotient rule gives the derivative of one function divided by another. Let

And require that g(x) not be zero. Then

In short,

**Definition of the Quotient Rule**

Find the derivative of the function

Solution

Here,

So,

As before, algebraic simplification is not always a good idea.

The chain rule is used in the case of a **function of a function**,
also known as a **composite function** or as a **composition of functions**. There
are two common notations with identical meanings:

As with linear combinations, products and quotients, composition
is another tool that allows us to build more complicated functions from simpler ones.
Compostions are particularly useful in the case where a given problem has so-called **chained
dependencies** where the quantity we wish to study is a function of a second quantity
which in turn depends on a third quantity. These chained dependencies arise in practical
applications so often they are given the special name **related rates** **problems**.

We start with two assumptions: g must be continuous and differentiable at x_{0}
and f must be continuous and differentiable at g(x_{0}). Using the slope form of
the definition of the derivative:

Now, if we define

Then by the continuity of g(x),

Put another way,

This allows us to make the substitution

Now we have

This is the * chain rule*:

**Definition
of the Chain Rule (function of a function version)**

Using the composition notation gives

**Definition of the
Chain Rule (composition version)**

Find the derivative of

Solution

We need to define the inside (g(x)) and outside (f(x)) functions:

Differentiating each gives

Now we apply the chain rule

Clearly, this example would have been very difficult without the chain rule - any attempt to multiply out the polynomial would take far too long and would be very likely to produce math errors.

- The classic calculus problem book - very light on theory, plenty of problems with full solutions, more problems with answers
- A simplified and updated version of the classic Schaum's Outline. Not as complete as the previous book, but enough for most students
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Schaum's Outline of Calculus (Schaum's...

Schaum's Easy Outline: Calculus