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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 2

Lesson 5: Applying the power rule

# Power rule

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.1 (EK)
Sal introduces the power rule, which tells us how to find the derivative of xⁿ. 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.
• At , Sal uses f'x to explain the Power Rule. Is f'x another way to write dx?