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Mth 271

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Power Rule for Derivatives

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In my independent studies I learned that applying the power rule, which states that for f(x) = x^n where n is NOT zero, f`(x)=nx^(n-1), to a given function will give us the derivative. In the notes you state that the derivative of a function in the form of f(x)= ax^2+bx+c is f`(x)=2ax+b.

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I see that these two statements are equivalent but I am wondering as to which you would prefer to see on a test. The power rule is a little easier for me and I see that it extends to exponential functions as well, where f(x)=ax^3, f`(x)=3ax^2.

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At this point of the course the power rule has not yet been introduced. We prove directly that the derivative is 2 a x + b, without reference to the power rule.

So if asked to prove this formula you would want to apply the limit definition to the quadratic function, rather than quoting the power rule.

An alternative on a question of this nature would be to prove the power rule, then apply it to the quadratic function. However to prove the power rule requires the formula for the binomial expansion and a more sophisticated argument. Naturally I would encourage this, but I certainly wouldn't require it at this stage of the course.

If the problem asked you to find the derivative of a function using the definition of the derivative, once more ou would be expected to apply the limit definition.

Now if a problem just asked you to find the derviative of some specific quadratic function, say 3 x^2 - 5 x + 9, you could either use the formula a x + b. or you could apply the power rule. Either would be acceptable.

Note that a x^3 is not an exponential function, it's a power function with power 3. An exponential function has the variable in the exponent, e.g., 3^x as opposed to x^3.

There are problems that ask you to find the derivative of a x^3 using the limit definition of the derivative. On other problems it would be fine to use the power rule.

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Is the power rule an appropriate application to be used on tests? Does it extend to other functions as well such as power or logarithmic functions?

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The power rule applies to any power function y = a x^p.

It does not apply to logarithmic or exponential functions. You'll see the rules for those functions a bit later. The rules are fairly simple. In fact, here's a preview:

The derivative of the natural log function ln(x) is 1 / x.

The derivative of the exponential function y = A b^x is A * b^x * ln(b). In particular the derivative of the basic exponential function y = A e^x is just A e^x, the same as the function itself.

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