Derivatives of Power Functions - Problem 1
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.