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### Course: Differential Calculus>Unit 2

Lesson 6: Power rule

# Justifying the power rule

​Proving the power rule for derivatives (only the more simple cases).
The power rule tells us how to find the derivative of any expression in the form ${x}^{n}$:
$\frac{d}{dx}\left[{x}^{n}\right]=n\cdot {x}^{n-1}$
The AP Calculus course doesn't require knowing the proof of this rule, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

## Let's first see whether the rule makes sense for $y=x$‍  and $y={x}^{2}$‍ .

Justifying the power ruleSee video transcript

## Now let's prove the rule for positive integer values of $n$‍ .

Proof of power rule for positive integer powersSee video transcript

## We can also prove the rule for $y=\sqrt{x}$‍ .

Proof of power rule for square root functionSee video transcript

## Want to join the conversation?

• In the first video, he graphed the derivative of f(x). Does this mean the derivative of a function is a function itself? And does this mean you can find the derivative of a derivative?
• Slow your roll here, not all functions are differentiable. The function sin(1/x^2) fails to be differentiable when x = 0. If the derivative exists, then it is a function - but not all functions are differentiable! Even more frustrating is that just because a function is differentiable once does not mean it's differentiable twice. In math we denote such functions as C^1 or C^2 - where the exponent indicates how many times it's differentiable, and the each one of those resulting functions is continuous. There are even C^infinity functions such as sin, cos, and e^x.
• In the second video, shouldn't the first term contain n choose 0 instead of n choose 1?
• You are correct, however, the n choose 1 coefficient in the proof is actually for the second term in the expansion. Sal just omitted the n choose 0 coefficient for the first (x^n) term since it just multiplies the term by 1. He possibly did this to make it more obvious that it cancels with the -x^n term in the numerator.
• In the first video, he graphed the derivative of f(x). Does this mean the derivative of a function is a function itself? And does this mean you can find the derivative of a derivative?
• Yes, the derivative of a function is also a function. As long as the derivative is a differentiable function, you may indeed find the derivative of a derivative.
• Shouldn't the value of f'(x) be undefined at x=0 for f(x)=x^x, because 0^0 is indeterminate?
• x^0 is 1 except 0 and the derivative is 1
• Is it bad that I don't know the Binomial Theorem yet? I haven't seen it covered yet in this course.
• where can we see the proof of ∫x dx=x2/2 + C?
• how do you differentiate a^mx
• To answer this you need the chain rule, which should be coming up soon ...
• What about non-positive integers. What if I want to take the derivative of a function with a negative or a fractional power?