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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!

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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.