Radical functions differentiation intro | Derivative rules | AP Calculus AB | Khan Academy

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·Nov 11, 2024

Let's say that we have a function f of x, and it is equal to -4 times the cube root of x. What we want to do is evaluate the derivative of our function when x is equal to 8. So, see if you can figure this out.

All right, now this might look foreign to you. You might say, "Well, I've never taken a derivative of a cube root before." But as we'll see, we can actually just apply the power rule here because this function can be rewritten.

So, f of x can be rewritten as -4. The cube root of x is the same thing as x to the 1/3 power. Now, it might be a little bit clearer that we can apply the power rule. We could take this 1/3, multiply it by this coefficient -4, so we have -4 times 1/3.

Now we have times x to the 1/3, and we just decrement that exponent. So, that's a different shade of blue to the 1/3 minus one power. This is the derivative. So, f prime of x is equal to that.

Now we just have to simplify. This is equal to -4/3 times x to the -2/3 power. If we want to evaluate f prime of 8, f prime of 8 is equal to -4/3 times 8 to the -2/3 power.

Well, that's the same thing as -4/3 times 8 to the 1/3 and then raised to the -2 power. I'm just using exponent properties here. If this looks completely unfamiliar, how I got from that to that, I encourage you to review exponent properties on Khan Academy.

Well, 8 to the 1/3, that is just 2. So, this is just 2, and then 2 to the -2 power. Remember, let me just take some steps here; it's a good review. This is equal to -4/3 times 2. The -2 is the same thing as 1 over 2 squared.

These two things are equivalent. 1 over 2 squared is the same thing as 2 to the -2. So, this is 1 over 4, and this is going to simplify to -4/12, which is equal to -1/3. And we are done.

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Radical functions differentiation intro | Derivative rules | AP Calculus AB | Khan Academy

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