Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

Question

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the derivative of Z with respect to T and the derivative of Z with respect to T when T equals 3 for the function Z equals T + 2 divided by T + 3. A says the derivative of Z with respect to T is 3 divided by T + 3 squad and the derivative when T E equals 3 is 1/12. E says there are 3 divided by T + 32 and 136. C 1 divided by T + 3 squared and 19th, and the D 1 divided by T + 32 and 136. No, we're given the function Z, OK? And we'll need to find the derivative and then the and then evaluate the derivative when T equals 3. So, first of all, how can we differentiate Z with respect to T? Well, notice that Z is a quotient, and we can differentiate it using the quotient rule. Recall that by the quotient rule it tells us that if we have a function y that's equal to you divided by v where you and V are. Functions of T, then the derivative of y with respect to T is going to be equal. The VDUDT minus UDVDT, OK. All divided by V2. So what this means is that we'll need to find the derivative of the numerator and the denominator in our quotient with respect to T and then use them in the quotient rule for differentiation to find the derivative of Z with respect to T, OK? So let's do that. Now in this case, if we let U be equal to T + 2, OK, and V be equal to T + 3. Then the derivative of U with respect to T is going to be equal to 1 because the derivative of T is 1 and the derivative of 2 is 0. While the derivative of V with respect to T is also going to be 1, OK, because we are differentiating T and a constant. Since we now have UV, and their derivatives, let's go ahead and use the quotient rule or apply them to the quotient rule. So now, that means the derivative of our function Z with respect to T. Is going to be equal to. Z, which is uh E + 3. Multiplied by one. Minus U which is T + 2 multiplied by DVDT which is also 1. I know all of this is going to be differentiated by V, which is T + 3 squared. Now, if we expand our brackets here, that's going to be T + 3 minus T minus 2 in our numerator, OK. And no, T minus T is 03 minus 2 is 1. So the derivative of Z with respect to T is 1 divided by T + 3 all squared. Now that we have the derivative of Z with respect to T, we can figure out its value at T equals 3. Because at T equals 3, then we can write D Z D T. To be equal to, well, let me. here. D Z D T and T equals 3 to be equal to 1 divided by 3 + 3 squared. 3 + 3 equals 6 and 62 equals 12 is 36. So the derivative of Z with respect to T when T equals 3 is 136. And like we said, the derivative of Z with respect to T generally is 1 divided by T + 3 squad. When we look back at our answer choices, that tells us D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

Key Concepts

Derivative

Quotient Rule

Evaluating Derivatives at a Point

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