Finding Derivative Values

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Question

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1

Accepted Answer

Welcome back everyone. Let f of U be equal to square root of u and U be equal to G of X, which is 2 x plus 3 divided by X minus 1. Find the derivative of the composite function F G X at X equals 2. For this problem we want to validate F of G of 2 derivative. So now, as we understand this is a composite function, which means that we need to apply the chain rule. First of all, we have to identify the derivative of F with respect to G at 0.2. And based on the chain rule we are multiplying by the derivative of the inner function. G at. 2. So now what does that mean? Well, essentially we need to evaluate each. Derivative, let's begin with G. We want to evaluate the G of X in the general case, and this means that we're differentiating to X + 3. Divided by x minus 1. Because we have the irrational function, we have to apply the quotient rule. So we take 2 x + 3 derivative. Multiply by x minus 1 and subtract 2 x + 3 multiplied by x minus 1 derivative. The derivative Of 2X plus 3 is going to give us 2. And the derivative of X minus 1 is going to give us 1. And we also need to include our denominator of X minus 1 squad based on the quotient rule. So now what we can do is basically distribute the terms, we got 2X minus 2. -2 X minus 3. Divided by x minus 1 squad. Combining the like terms, we get -5. Divided by x minus 12. So we have our derivative and now we can evaluate G at 2, which is -5. Divided by 2 minus 1 square. And that's -5 divided by 1, which is -5. We also need to evaluate F with respect to G, right? We know that F of u is equal to square root of u. Which is essentially F of G of X. So now if we differentiate off of you. We're going to get our derivative in a general case. That's the derivative of square root of you. Which is the derivative of U to the power of 1/2, and that's 1/2 EU to the power of -12 using the power rule. This is equal to 1 divided by 2 square roots of U. So we have our derivative in terms of you, right? And essentially what we have to do. Is substitute for you so that we can get it in terms of X. We're going to get 1 divided by 2 square roots of. Substituting for you gives us. 2 X plus 3 divided by X minus 1. So now we have our derivative in terms of X, and we can evaluate F at G of 2. Which means that we're going to replace X with 2. And this gives us 1 divided by 2 square roots of. 2 multiplied by 2. Plus 3 divided by 2 minus 1. Which is 1 divided by 2 square roots of 7. And now we are nearly done. We are going to substitute our result for F at G of 2, which is 1 divided by 2 square root of 7. And multiplied by G at 0.2, which is -5. This gives us -5 divided by 2 square roots of 7. And rationalizing the denominator, we can also write that this is -5 square root of 7 divided by 2 multiplied by 7, which is 14. And that's our final answer for this problem. Thank you for watching.

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1

Key Concepts

Chain Rule

Composite Functions

Evaluating Derivatives

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