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Derivatives of Power Functions - Problem 1 4,401 views

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The formula for finding the derivative of a power function f(x)=xn is f'(x)=nx(n-1). For example, if f(x)=x3, then f'(x)=3x2. When a power function has a coefficient, n and this coefficient are multiplied together when finding the derivative. If g(x)=4x2, then g'(x) = 2*4x1=8x. Radical functions, or functions with square roots, are also power functions. The square root of x, instance, is the same as x(1/2). When finding the derivative, once more, use the formula: the derivative of x(1/2) is (1/2)x(1/2-1)=(1/2)x(-1/2). Recall that power functions with negative exponents are the same as dividing by a power function with a positive exponent. One example of this is h(x)=x(-5)=1/(x5). To find the derivative of a function with negative exponents, simply use the formula: h'(x)=-5x(-5-1)=-5x-6=-5/(x6). It is important to remember that you are subtracting 1 from the power n.

We're talking about the derivative of power functions. We have the formula here the derivative with respect to x, of x to the n, is n times x to the n minus 1. So it's a really easy formula.

Let's start with an easy example. Let's compute the derivative with respect to x of x². So that's going to be this exponent 2 is going to come out in front 2x. Then I replace it with 1 less. 1 less than 2 is 1, so it's 2x. Easy. How about this one? X to the 15. The derivative will be 15x to the 14.

Now here it's important to recognize power functions even when they're not written in the form of a power function. This is the same as the derivative with respect to x of x to the -2. So you can still apply the power function rule here. The -2 comes out in front, and you have x2 one less. Now one less than -2 is -3. So this is the same as let me write over here -2 over x³.

Finally, the derivative with respect to x. You can also use this for radical functions, because these are also power functions. This is the same as the derivative with respect to x of x to the 1/3.

So you take the power, 1/3 that comes out in front. You replace it with one less. Now one less than 1/3 is -2/3. You can simplify that. If your teacher doesn't like negative exponents, or they don't like fractional exponents, you can write it as 1/3, 1 over x to the 2/3. If they don't like fractional exponents remember that x to the 2/3 is the same as 1 over 3 cube root of x².

So remember the derivative of a power function is nx to the n minus 1, where n is the original exponent.

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