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) = x^n, where n is any constant, then f'(x) = nx^{n-1}. Equivalently, the derivative of x^n is nx^{n-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.

    If f(x) = , then f'(x) =       Pressing "Peek" will decrease your health!
    If f(x) = , then f'(x) =      
    If f(x) = , then f'(x) =      
    If f(x) = , then f'(x) =      
    If f(x) = , then f'(x) =      
    If f(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

In each of the following, give the answer in rational form as in the above table; that is, without negative exponents. For instance, write -3x^{-4} as \frac{-3}{x^4}.

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
    [x^3] = 3x^2
      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.
In other words, to find the derivative of a sum (or difference) of several function, just find the derivative of each function and add (or subtract).

The derivative of c times a function is c times the derivative of the function.
In other words, to find the derivative of a constant times a function, just find the derivative of the function and multiply by the constant.

 
Quick Examples

\frac{d}{dx}[1 + x^3] = 0 + 3x^2 = 3x^2   Property (A)
\frac{d}{dx}[x^2 - x^3 + x^5] = 2x - 3x^2 + 5x^4 Property (A) works for three or more terms
\frac{d}{dx}[4x^3] = (4)3x^2 = 12x^2 Property (B). In effect, multiply the exponent by the coefficient
\frac{d}{dx}[12] = \frac{d}{dx}[(12)(1)] = (12)0 = 0 Because the derivative of 1 is zero.
So, the derivative of any constant is zero.
\frac{d}{dx}\Bigleft[\frac{4}{x}\Bigright] = \frac{d}{dx}\Bigleft[(4)\frac{1}{x}\Bigright] = (4)\frac{-1}{x^2} = \frac{-4}{x^2} Property (B) again
\frac{d}{dx}[3x^3-4x+7] = (3)3x^2 - (4)1 + (7)0 = 9x^2-4 Combining the properties

Some for You

    \frac{d}{dx}[ ] =      
    \frac{d}{dx}[ ] =      
    \frac{d}{dx}[ ] =      
    \frac{d}{dx}[ ] =      
    \frac{d}{dx}\Bigleft[ \Bigright] =      
    \frac{d}{dx}\Bigleft[ \Bigright] =      
    \frac{d}{dx}\Bigleft[ \Bigright] =      

 
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.

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Last Updated: October 2009
Copyright © 2008 Stefan Waner