Derivatives of Powers, Sums and Constant Multiples
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: Some for You: Enter the required expressions using valid technology format. 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,
f(x) | = | x4 | = | x-4 | implies | f'(x) | = | -4x-5 | = | x5 |
This allows us to expand the above table a little:
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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 derivative of x3 with respect to x equals 3x2."
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
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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
(B) [cf(x)]' = cf'(x). In differential notation, these rules are
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.
Some for You |
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.