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: Some for You: Enter the required expressions using valid technology format. Want to see a proof of the power rule? Click here. |
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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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.
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!).