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.

b. ƒ(x)g²(x), x = 0

Accepted Answer

Welcome back once to another video. Find the derivative of F2X G of X when X is equals 2 using the table given below. So first of all, let's identify the general expression of the derivative of the composite function we have F2d of X multiplied by G of X. We're going to differentiate this expression and we have to recall the product rule, right? So first of all, we're taking the derivative of the first function, which is the derivative of F2 X. We're multiplying by the second function which is G of X, and then we are adding the first function F2 X. Multiplied by the derivative of the second function, so G of X, and as we can see, we need to identify the derivative of F squad of X, right? So it is a composite function as well. We first will apply the power rule which gives us 2 F of X, and then we are multiplying by the derivative of F of X according to the chain rule, right, because it is a composite function. This is then multiplied by G of X in our original expression, and we are adding F2X multiplied by G of X. Well done. So we have our general expression. And now if we want to identify the derivative of F2X multiplied by G of X. At x equals 2. We simply need to replace every X with 2, so we get 2 of 2 multiplied by F.2 multiplied by G at.2 plus F squared at 0.2 multiplied by G at 0.2. And this is where we need to use our table, right? So first of all, we need to identify F at.2. So we identify F of X in the 2nd row, and if X is equal to 2, then F is equal to 1. Now we need F at 0.2 so we identify F of X at X equals 2, it is equal to 13. Then we need G at. 2. So let's take G of X from the table and identify the intersection with X equals 2. This gives us 5, right? Then we need F of 2. We already know that it is equal to 1. We will just need to square it later, and then we need a G at 0.2. So G of X at X equals 2 is equal to 6 according to the table. So now we have all of the numerical values that we need. Let's substitute them. We have 2 multiplied by 1, multiplied by 13, multiplied by 5. Plus 1 squared multiplied by 6. So let's perform the calculations. 2 multiplied by 1 is 22 multiplied by 13 is 26, and 26 multiplied by 5 is 130. So we end up with 130 plus 1 multiplied by 6, which is 6. So overall our sum is 136, which corresponds to the 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.

b. ƒ(x)g²(x), x = 0

Key Concepts

Product Rule

Chain Rule

Evaluating Derivatives at Specific Points

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