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### Course: AP®︎ Calculus AB (2017 edition)>Unit 3

Lesson 2: Power rule

# Power rule

Let's dive into the power rule, a handy tool for finding the derivative of xⁿ. This rule simplifies the process of taking derivatives, especially for polynomials, by bringing the exponent out front and decrementing the power. We explore examples with positive, negative, and fractional exponents. Created by Sal Khan.

## Want to join the conversation?

• Does the power rule tell us how to deal with an expression with a coefficient, like f(x) = 2x^3? And what do I do if I have more than one term in my equation, like f(x) = 3x^2 + x + 3?
• What would the derivative be of something like 2^X? (two raised to the power of x) Would it still be 2x?
• No.

The Power Rule is for taking the derivatives of polynomials, i.e. (4x^5 + 2x^3 + 3x^2 + 5). All the terms in polynomials are raised to integers.

2^x is an exponential function not a polynomial.

The derivate of 2^x is ln(2)*2^x, which you would solve by applying the Derivative of Exponential Rule: The derivative of an exponential function with a base of C is the natural log of C times the exponential function.

Derivate of C^x = ln(C) * C^x
In this case, C = 2. So... derivate of 2^x = ln(2) * 2^x

Sal does a a proof for common functions, in one of the later tutorials that probably walks you through a rigorous proof of it. (I haven't seen it yet).
• Why can't n = 0? If n is 0, then x^n is 1, right? Then its derivative is 0 like any other constant. And that follows the power rule doesn't it? So why can't n = 0?
• There is no reason why n can't be 0 for the power rule in differentiation. The reason why we say this is because this is more convenient when we reverse the power rule when calculating antiderivatives. For example, if I told you dy/dx=6x^2, with the power rule reversed we can show that y=2x^3. This is not possible with dy/dx=1/x, as we would be dividing by zero.

For differentiation, n can be 0. There is nothing wrong with it.
• At , can n be imaginary?
• Yes, it can. But, of course, working with complex exponents is a bit difficult, although the power rule still applies.
Thus,
d/dx 5x^(3i) = 15𝑖x^(-1+3𝑖)
• Is there ever a case where you take a derivative twice?
So for example, x^3-5x^2+12.
Using the power rule, you'd get 3x^2-10x
Is there a case where you would apply the power rule again and get 6x-10?
• Does the power rule work If you have a function with a square root or a fraction?
• Yes. For example, if you have square root of x as f(x) which is x^1/2, you use the power rule to get 1/2*x^-1/2 which is just 1 divided by (2*square root of x). Sorry, I don't know how to notate square roots on KA, if it's even possible.
• The test questions to this lesson include the following example:
a= 2x^4 + 6x^3 - 7x^2
and gives the answer (a prime) = -14x
Why would the answer not also include the terms: 8x^3 + 18x^2?
• I'm not sure why because a' should equal 8*x^3+18*x^2-14*x. What does a mean in this problem's case. Is it referring t f(x) or what?
• When asked for a proof of a derivative of a constant, can the power rule be used as where c represents the constant and x the variable:

y=cx^0
y'=c*0*x^-1
Because there the entire term is multiplied by zero, the expression for the derivative is equal to 0?
• I think that would be adequate, but you can more directly prove it from the definition of a derivative: since for a constant f(x) is the same value for all x , you have a 0 in the numerator. Thus, the derivative of all constants must be 0.
• How do I figure out if a number can be expressed as a power greater than 1? Example 120, 400, 100, 250, 200 and how do you express the number as a power?
• You could factor each number and look for factors that occurs an even number of times.

100 = 2 * 2 * 5 * 5 = (2 * 5) * (2 * 5) = 10 * 10 = 10^2
400 = 2 * 2 * 2 * 2 * 5 * 5 = (2 * 2 * 5) * (2 * 2 * 5) = 20 * 20 = 20^2

Some numbers will give you several options, e.g. 81 = 3^4 = 9^2.