Derivatives of Powers, Sums and Constant Multiples

This topic is also in Section 4.1 in Applied Calculus 5e or Section 11.1 in Finite Mathematics and Applied Calculus 5e

Q Computing the derivative of a function is a pretty long-winded process. Isn't there an easier method?
A For practically all the functions you have seen, the short answer is "yes". In this section we study short-cuts that will allow you to write down the derivative powers of x (including fractional and negative powers) as well as sums and constant multiples of powers of x, such as polynomials. We start with the rule that gives the derivative of a power of x:

Power Rule

If f(x) = xn, where n is any constant, then f'(x) = nxn-1. Equivalently, the derivative of xn is nxn-1.

Quick Examples

The following table shows several examples of derivatives of powers of x:

f(x)
1
x
x2
x3
xn
f'(x)
0 
1 
2x 
3x2
nxn-1

Some for You: Enter the required expressions using valid technology format.

Q If f(x) = x4, then  f'(x) =      

Q If f(x)= x, then f'(x) =      

Want to see a proof of the power rule? Click here.

Negative Exponents

Since the power rule works for negative exponents, we have, for example,

This allows us to expand the above table a little:

f(x)
xn
1
x
x2
x3
1

x
1

x2
1

x3
f'(x)
nxn-1
0
1
2x
3x2
-1

x2
-2

x3
-3

x4

 

Differential Notation

Differential notation is based on an abbreviation for the phrase "the derivative with respect to x." For example, we learned above that if f(x) = x3, then f'(x) = 3x2. When we say "f'(x) = 3x2," we mean:

The phrase "with respect to x" tells us that the variable of the function is x and not some other variable. We abbreviate the phrase "the derivative with respect to x" by the symbol "d/dx."
 

Derivative With Respect to x

The notation
d

dx
means the derivative with respect to x. Thus, for instance,

    d

    dx
    [x3] = 3x2
      The derivative, with respect to x, of x3 equals 3x2
    d

    dx
    [1] = 0
      The derivative, with respect to x, of 1 equals 0
    d

    dx
    1

    x
    =
    -1

    x2
      The derivative, with respect to x, of 1/x equals -1/x2

 
We can find the derivative of more complicated expressions using the following:
 

Derivatives of Sums, Differences and Constant Multiples

If f'(x) and g'(x) exist, and c is a constant, then

    (A) [f(x) ± g(x)]' = f'(x) ± g'(x),

    (B) [cf(x)]' = cf'(x).

In differential notation, these rules are

    (A)
    d

    dx
    [f(x) ± g(x)] =
    d

    dx
    [f(x)]±
    d

    dx
    [g(x)]
    (B)
    d

    dx
    c[f(x)] = c.
    d

    dx
    [f(x)]

In words:

The derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives.

The derivative of c times a function is c times the derivative of the function.

 
Quick Example

    d

    dx
    [3x3 + 4x - 9] = 3(3x2) + 4(1) - 0
    = 9x2 + 4

Some for You

Q
d

dx
[4x1.2 - 2x - 9x-1] =      


Q
d

dx
4x0.2 + 9.6 -
2

x3
=      

 
Try some exercises for this topic here. Alternatively, try some of the exercises in Section 4.1 in Applied Calculus or Section 11.1 in Finite Mathematics and Applied Calculus.

Alternatively, press "game version" on the sidebar to go to the game version of this tutorial (it has different examples to try and is a lot of fun!).

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Last Updated: October 2009
Copyright © 2007 Stefan Waner