Lecture Note
University
Rice UniversityCourse
Preparing for the AP Calculus AB ExamPages
2
Academic year
2022
Awayne
Views
13
Power Rule for Derivatives The power rule is one of a number of derivative rules in calculus. The power rule says thatif a function is raised to a power, then the derivative is equal to that function raised to thesame power times the original function's derivative.For example, suppose we are given a function ƒ(x) and its derivative ƒ'(x), then the powerrule says:If ƒ(x) = xn, thenƒ′(x) = nxn-1 The power rule is a shortcut for computing derivatives of exponents. So it's possible that you have encountered the fact that if you were to define the derivativefor every single derivative that existed for square functions, you would be in for a very longyear. But what if I said, "hey let's take the derivative of x to the 10th power"? Of course, there is Pascal's triangle and the rules for multiplying out a binomial. However,that is a time-consuming process that is not worthwhile. There are also trig functions andexponential functions, as well as logarithmic functions—so many functions that it would bemore efficient to take their derivative than to use them directly. So instead of trying to solve the problem in its current form, we'll take a different approach.We'll take the derivative of the function three and graph it. The derivative of a horizontal line at any point on the line is 0. The derivative of x is theslope of a horizontal line tangent to the graph at that point. Okay, so now let's take the derivative of x. The slope of a line that is tangent to the graphat a point is equal to the derivative of the function at that point. If I draw such a line, it hasa slope of 1. You might have seen this from the example video. We have already done 3x squared,which gave us 6x, hint. Yes, the derivative of x squared is 2x. The slope of a line is always changing from positive to negative and back again. The slopeof the line y=2x can be found at the point where its value is 0, which is where x=0. So what is the pattern? What is the trick? Can you solve for the derivative of x cubed? You are correct, the derivative of x4 is 3x2. A pattern can be difficult to see at first, but if we examine it closely, we can see that theexponent becomes the coefficient, and then we subtract 1 from the exponent. So thismeans we have 4x cubed. If I was to formulate a rule, it would be called the power rule for derivatives. The power rule for derivatives states that if x is raised to the nth power, then the derivativeof x with respect to t equals nxn-1. For any exponent, even rational expressions, the square root function will work. Forexample. If I have 1 over x squared, that can be rewritten as x to the negative 2. Thesquare root of x is x to the 1/2.
If the function could be written as a power, I could take its derivative by using this rule. Ifthere were a constant coefficient in front of the x, like 3x2, then I could take the derivativeby multiplying that constant by itself. In our example, we saw this phenomenon: if you take the derivative of x2, you can factorout a 3. The derivative of x2 is 2x times 3, which gives us 6x.
Power Rule for Derivatives
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