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## AP®︎ Calculus AB (2017 edition)

### Unit 3: Lesson 4

Derivatives of negative and fractional powers with power rule- Differentiating integer powers (mixed positive and negative)
- Power rule (negative & fractional powers)
- Fractional powers differentiation
- Power rule (with rewriting the expression)
- Radical functions differentiation intro
- Differentiate integer powers (mixed positive and negative)
- Worked example: Tangent to the graph of 1/x
- Power rule review

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# Fractional powers differentiation

Sal differentiates h(x)=5x^¼+7, and evaluates the derivative at x=16. This can actually be done quite easily using the Power rule!

## Want to join the conversation?

- Why do you cube 16^(1/4)?(8 votes)
- According to exponent rules,
**x^(ab) = (x^a)^b**. Therefore,**16^(3/4) = 16^(1/4*3) = (16^(1/4))^3**.(2 votes)

- At02:11how does Sal know that 16^(1/4) is two. I thought it's 4 in the very first moment ( i know 4 is wrong^^). Should one really be able to do such things with the brain or should one use a trick for calculating it..(3 votes)
- It's quite simple to be honest. Since its one over some 2^n in which n ∈ natural numbers. This means that we're dealing with square roots. If we rearrange it to 16^(1/(2^2)) (which looks complex but if you write it down its not) then we can see that we're just taking the square root two times. So square root of 16 is 4 then square root of 4 is 2(5 votes)

- How do you read the fractional powers?(0 votes)
- So, using the function shown above as an example, we can say that h(x) = 5(x^(1/4)) + 7 is "h of x is equal to five times x (raised) to the one-fourth (power) plus seven"...

Both raised and power are optional... Generally, we don't use them but sometimes, when the expression in question is too complex, using the above terms makes it more readable and easier to read...

There are many other ways of reading the fractional powers but I guess, the one I mentioned is most commonly used...(0 votes)

## Video transcript

- [Voiceover] So we have h of x is equal to five x to the
1/4th power plus seven. And we wanna find what is h prime of 16, or what is the derivative of this function when x is equal to 16? And like always, pause this video and see if you can figure
it out on your own. All right, well let's just take the derivative of both sides of this. So on the left hand side,
I'm gonna have h prime of x, and on the right hand side, well the derivative of
the right hand side, I can just take the derivative
of five x to the 1/4th and add that to the derivative, with respect to x of seven. So the derivative of five
x to the 1/4th power, well, I can just apply
the power rule here. You might say, wait, wait wait, there's a fractional exponent, and I would just say, that's okay. The power rule is very powerful. So we can multiply the
1/4th times the coefficient. So you have five times 1/4th x to the 1/4th minus one power. That's the derivative of
five x to the 1/4th power. And then we have plus seven. Now what's the derivative of seven, with respect to x? Well seven doesn't
change with respect to x. The derivative of a constant, we've seen this multiple
times, is just zero. So it's just plus zero. And now we just have to simplify this. So this is gonna be h prime of x is equal to 5/4ths x to the, what's 1/4th minus one? Well that's negative 3/4ths. That's 1/4th minus 4/4ths,
or negative 3/4ths. So 5/4ths x to the
negative 3/4ths plus zero, so we don't have to write that. And now let's see if we can evaluate this when x is equal to 16. So, h prime of 16 is 5/4ths times 16 to the negative 3/4ths. Well that's the same thing as 5/4ths times one over 16 to the 3/4ths, which is the same thing
as 5/4ths times one over, let's see, I could view this as 16 to the 1/4th, and then cubing that. And so, what is this? 16 to the 1/4th is two, and then you cube two. Two to the third power is
eight, so that's eight. So you have 5/4ths times 1/8th, which is going to be equal to five times one is five, and then four times eight is... four times eight is 32. And we are done.