21–30. Derivatives
b. Evaluate f'(a) for the given ...

Question

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 1/√t; a=9, 1/4

Accepted Answer

Welcome back, everyone. In this problem, given the function G of X equals 1 divided by X to the third, find G B for B equals 8 and B equals 127. A says G 8 is -196, while G of 127 is -81. B says it's -148 and 27. C -196 and 27. And a D in -148 and the 81. Now, if we're going to find the G B for different values of B, we'll need to find the G of X. In other words, the derivative of X. So let's the derivative of G of X. sorry. So let's differentiate G of X and then see if we can solve for those values of B, OK? So far we know that geo X is equal to 1 divided by X 1/3. And we could rewrite this as x to the power of -1/3 based on what we know about powers. When we do that, then to differentiate G of X to get G of X, we can apply the power rule which tells us that the derivative. It's right over here. Of X rays to our power is equal to the power multiplied by X raises to the power of the well, it is to the power of N minus 1, where N represents the power, OK. So applying the poor rule here, that means G of X is going to be equal to -1/3, multiplied by X, raised to the power of -1/3 minus 1. Now, when we subtract 1 from -1/3. We are left with X to the pore of -4/3. So G of X is equal to -13 X to the pore of -4/3, or we could even rewrite this as -1 divided by 3 X raised to the pore of 4/3. Now that we have an expression for G of X, let's find G B for both the values. OK. Now, when B is equal to 8, OK. That means G 8 is going to be equal to -1 divided by 3 multiplied by 8, raised to the core of 4/3. Now we could rewrite it to the poor of 4/3 as it raised to the power of 1/4. Again, based on what we know about powers, we can rewrite their product as raising one power to the other. Now the reason why we wrote this with 8 to the power of 1/3 is because we know that equals 2. So this is going to be -1 divided by 3 multiplied by 2 to the 4th, 2 to the 4th is 16. And 3 multiplied by 16 is 48. So G 8 is -148. Now that we have G 8. Next, let's move on to G of 127. That is when B equals 127. I know to solve this one, OK. I'm going to use the form of for derivative, G X where it's -1/3 X to the negative 4/3. So, sorry, this should be 127. So G of 1/27 is going to be equal to -13 multiplied by 1/27 raised to the power of -4/3. Now why did I write it like this? Well, we could rewrite 1/27. As 27 to the power of -1, OK? I know based on what we know about powers. We could multiply our pores here, -1 multiplied by -4/3, and rewrite this as -1/3 multiplied by 27 to the power of 4/3, 4/3. We could take this a step further, OK? And rewrite this as 27 to 3, raise to the power of 4, and we did that conveniently because 27 to 3 power would be the cube root of 27, which is 3. So now we have -13 multiplied by 3 to the 4th. In this case, OK, we could rewrite -1. Well, we know -13 is going to cancel one of the powers here for 3 to 4th, so we could rewrite this as -3 to the 3rd and 333 is 333 is 27. Therefore, G of 127 is equal to -27. OK. And G 8 is -148. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 1/√t; a=9, 1/4

Key Concepts

Derivatives

Chain Rule

Evaluating Derivatives at Specific Points

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