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:
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Power Rule
If f(x) = xn, where n is any fixed real number, then f'(x) = nxn-1. Equivalently, the derivative of xn is nxn-1. Quick Examples Some for You (Note: you will need to enter an algebraic formula 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
| f(x) | = |  x4 | = | x-4 |
| f'(x) | = | -4x-5 | = | x5 |
This allows us to expand the above table a little:
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Differential Notation is This topic is also in 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 nothing else. We abbreviate the phrase "the derivative with respect to x" by the symbol "d/dx."
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Derivative With Respect to x
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We can find the derivative of more complicated expressions using the following:
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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 |
Now try some of the exercises in Section 3.7 in Applied Calculus or Section 10.7 in Finite Mathematics and Applied Calculus.
