In Exercises 19–22, find the values of the derivatives.
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Question

In Exercises 19–22, find the values of the derivatives.

dr/dθ |θ₌₀ if r = 2/√(4 – θ)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says using the limit definition, find the value of the derivative, and we're given the derivative of S with respect to theta evaluated at theta equal to 8. If S is equal to 3 divided by the square root of the quantity of 9 minus data in quantity. We're given 4 possible choices as our answers. For choice A, we have 2 divided by 3. For choice B, we have 3 divided by 2. For choice C, we have 5 divided by 3, and for choice D, we have 7 divided by 4. So we need to find the value of the derivative given in the problem using the limit definition of a derivative. So we need to find the derivative of S with respect to theta, evaluated at theta equal to 8. Using the limit definition. So this derivative is going to be equal to, and here we call your limit definition of a derivative, that's going to be the limit, as H approaches 0, of the quantity of S of 8 plus H minus S of 8 in quantity, divided by H. So, substituting in the expression for S for the two terms of the numerator, this is going to be equal to the limit. As H approaches 0, of the quantity for S of 8 plus H, we're going to have 3 divided by square root of the quantity of 9 minus the quantity of 8 plus H. In quantity Then we have minus S of 8, which is 3 divided by the square root of the quantity of 9 minus 8 in quantity, and all that is still being divided by H. So simplifying our two terms in the numerator, this is going to be equal to the limit, as H approaches 0 of 3, divided by the square root of the quantity of 1 minus h in quantity, minus 3 in quantity divided by h. So if we try to evaluate this limit by direct substitution at this point, we would get 0 divided by 0, which is an indeterminant form. So we're going to need to rearrange our expression here in order to take the limit. So the first thing we can do is factor out 3 out of both terms in our numerator, and then combine what's left using a common denominator. So doing so, this is going to be equal to the limit, as H approaches 0 of 3 divided by H, multiplying the quantity of 1 minus. The square root of the quantity of 1 minus h in quantity, and that quantity is divided by the square root of the quantity of 1 minus H in quantity. Now we want to remove the radical of the numerator, and we can do this by multiplying by 1 in the form of the conjugate of the numerator. So we'll multiply this by the quantity of 1 plus the square root of the quantity of 1 minus h in quantity divided by. The quantity of 1 plus the square root of the quantity of 1 minus h in quantity. So, multiplying everything out in this expression, this is going to be equal to the limit. As h approaches 0 of 3 divided by H, multiplying the quantity of 1 minus the quantity of 1 minus H in quantity, that quantity is divided by the quantity of the square root of the quantity of 1 minus h in quantity plus 1 minus h in quantity. So simplifying this expression, this is going to be equal to the limit, as H approaches 0. Of 3 multiplied by H, since the quantity of 1 minus the quantity of 1 minus H in quantity is just equal to H, the ones cancel out, and then we have minus minus H, which gives us just 8. So our numerator here is 3H, and that gets divided by everything in the denominator, which is H, multiplying the quantity of the square root of the quantity of 1 minus H in quantity, plus 1 minus H in quantity. The h in the denominator will cancel out with the h in the numerator, and now we can evaluate this limit by direct substitution. So this is going to be equal to 3 divided by the quantity of the square root of the quantity of 1 minus 0 in quantity plus 1 minus 0 in quantity. And evaluating this expression, this is going to be equal to 3 divided by 2. So this is the answer that we are looking for for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

In Exercises 19–22, find the values of the derivatives.

dr/dθ |θ₌₀ if r = 2/√(4 – θ)

Key Concepts

Derivatives

Chain Rule

Evaluating Derivatives at a Point

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