Suppose that functions ƒ(x) and g(x) and their first ...

Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

c. ƒ(x) , x = 1

g(x) + 1

Accepted Answer

Welcome back, everyone, to another video. Find the derivative of G of X divided by F of X minus 4, when X is equal to 2 using the table given below. So first of all, let's identify the derivative of G of X divided by F of X minus 4. And because we have a rational expression, we're going to apply the quotient rule. So we have to recall that. First of all, we are taking the derivative of the top function, multiplying it by the bottom function, which is F of X minus 4. And then we have to subtract the top function G of X, which is multiplied by the derivative of the bottom function. So F of X. Minus 4, the derivative of 4, right? And we are dividing by the square of the bottom function F of X minus 4 squad. So now let's simplify. We're going to get GX multiplied by F of X minus 4 and then we are subtracting G of X, which is multiplied by FX since the derivative of 4 is 0 and all of that is divided by F of X. Minus 42. So now what we're going to do is simply evaluate this result at X equals 2, right? So we can say that G of X divided by F of X minus 4 derivative. At X equals 2 is equal to. G at 2 because now our axis 2 multiplied by F at. 2 minus 4 minus G at. 2 multiplied by F at 2 and all of that. Is divided by F at 0.2 minus 4 squad. So now we can clearly see what terms we need to use from the table. The first one is G at.2. So we identify G, we identify X equals 2, and we can see that it is equal to 6, right? So that term is equal to 6. Now we need F at 0.2. We identify F of X. At X equals 2, it is equal to 1, so we can say that this term is equal to 1. We also need G at.2, so we identify G of X, at X equals 2, it is equal to 5. We also need F at. 2, so we identify F at X equals 2, according to the table, it is equal to 13. And now in the denominator we only have F of G, which is 1, right? We already know that. So now let's substitute all of those values. We end up with 6 multiplied by 1 minus 4, minus 5 multiplied by 13, and this is divided by 1 minus 4 squared. Well then, we have all of the numerical expressions. First of all, we have 6 multiplied by -3, which is -18 minus 65. Right, 5 multiplied by 13, and this is divided. Why 1 minus 4 squad, so -3 squad is simply 9. So let's simplify, -18 minus 65 gives us -83. Divided by 9. And that's our final answer to this problem. Thank you for watching.

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

c. ƒ(x) , x = 1
g(x) + 1

Key Concepts

Derivative of a Function

Sum Rule of Derivatives

Evaluating Derivatives at Specific Points

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