   # Derivative of Sum of Monomials

 Our Introductory Calculus tutorial software program contains over 40 topic areas. One of them is Derivative of Sum of Monomials, and this page summarizes the main ideas of this topic. This page is intended for review, and is not a substitute for the interactive, self-paced tutorials of the MathTutor introductory calculus software program.

General Explanation

In this lesson we will learn how to differentiate a sum of monomials,
such as u(x) + v(x). First, let's derive the rule.

We'll begin with the definition of a derivative:

```dy        f(x+Δx) - f(x)   = lim                dx   Δx→0       Δx
```

Now, let f(x) = u(x) + v(x). Then, f(x+Δx) = u(x+Δx) + v(x+Δx).

```dy        u(x+Δx)+v(x+Δx) - [u(x)+v(x)]   = lim                               dx   Δx→0             Δx
```

Now let's distribute the minus sign and switch the positions of the second and third terms.

```dy        u(x+Δx)-u(x) + v(x+Δx)-v(x)   = lim                             dx   Δx→0             Δx

u(x+Δx)-u(x)        v(x+Δx)-v(x)   = lim               + lim                   Δx→0       Δx       Δx→0       Δx
```

We can see that this is actually the sum of du(x)/dx and dv(x)/dx.

As a general rule:

```d(u(x)+v(x))   d(u(x))   d((v)x))             =         +              dx          dx         dx
```

Sample Problem

Find:

```d(6x2 + 9x6)                 dx
```

Solution

We will use the general rule for the derivative of the sum of monomials.

```d(6x2+9x6)    d(6x2) + d(9x6)           =                    dx         dx       dx

= 6•2•x2-1 + 9•6•x6-1 = 12x + 54x5
```