Suppose that functions f and g and their derivatives with resp...

Question

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

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Find the derivatives with respect to x of the following combinations at the given value of x.

h. √(f²(x) + g²(x)), x = 2

Accepted Answer

Welcome back, everyone. Given that G of -2 equals 5, G-2 equals -4, H-2 equals 2, and H-2 equals 3, find the derivative of the square root of 4G X2 + 11 H of X2 at X equals -2. A says it's 7/6, B 76, C 7/12, and D negative 712. Now there are two parts to this problem. First, we want to find the derivative of our expression, and then we want to evaluate it at X equals -2. How can we differentiate this expression? Well, notice that we have an expression under a square root, OK? So that means at some point we'll need to use the chain rule and recall that the chain rule of differentiation basically says that if we are differentiating a composite function F of G of X. Then the derivative is equal to the derivative of the other function F of G of X, multiplied by the derivative of the inner function G of X. Now, we also know here, OK, that since this function is under a square root, it would also be good to recall the derivative of another function that looks like that. For example, the derivative of the square root of X, which would be equal to 1 divided by 2 root X. So no, that means if we apply that, the derivative of the derivative. Of the square root. Of 4 geopic squared. Plus 11 H of X squared, yeah. Is going to be equal to 1 divided by 2 multiplied by the expression root 4 g of squared. Plus 11 H of X squared. OK. So that's the derivative of the alter function multiplied by the derivative of the inner function, uh, the expression inside our square root, OK? It's 4G of X2 plus 11 H of X squared. So no, to solve, all we need to do is to find the derivative of uh this expression. Now if you think about it here, let me move on to this line, OK? Uh we're gonna essentially write back our first term in this expression, OK. But now what will be the derivative of the second of of our inner function? Well, here it's remember it's 4 multiplied by G of X squared. So we can differentiate this using the uh the power rule of differentiation. So we write back 4. Now we bring down the power it's 4 multiplied by 2 GFX rays to the power of 2 minus 1 because we subtract 1 from the power multiplied by the derivative of GFX, OK, because we're treating it as a uh composite function. We're squaring the function plus. 11 multiplied by 2 to 2 minus 1 of X multiplied by the derivative of H of X, OK. We treat it the same way that we did with our function G. No, when we go ahead and simplify here, that means our function is going to be 4 geopics, OK, multiplied by. Uh, the derivative of GFX. OK, because here it's still 4 because 2 cancers too. There's 4 G of X multiplied by the derivative of G of X plus 11 HX multiplied by the derivative of HF X. Again, 2 was canceled because we factored it out of our expression. I know all of this is gonna be divided by the square root. Of 4 G of X2 + 11 of X2. All we need to do now is to substitute the values that we know. Remember that at X equals 2 or at X equals -2, we already know that G-2 E equals 5, as we stated earlier, G-2 E equals -4. H Uh, sorry, H-2 equals 2 and H of -2 equals 3. So we have all of this information already, OK? So now, in that case, at X equals -2, our expression is going to become 4G of -2 multiplied by G of -2 plus 11 H of negative2 multiplied by H-2. All divided by the square root of 4G -2 squad. Plus 11 multiplied by H of -2 squad. And since we know what those values are, then it is gonna become 4 multiplied by G of -25 multiplied by G of -2 -4 plus 11 multiplied by H-22 delied by H-23. All divided, OK, by the square root, OK, 4 multiplied by G of -22, so that's 5 squared, plus 11 multiplied by H of -22. So that's multiplied by 22, OK? Now, when we go ahead and simplify, let me make some space to the right here, OK. So that's 4 multiplied by 5 multiplied by -4, -80. 11 multiplied by 2 by 3, that's 66. And this is all divided by 4 multiplied by 52, which is 25, that's 100. Plus 11 multiplied by 2 square, that's 11 multiplied by 4, which is 44. 80 + 60 is -14. 1 100 + 44 is 144, and the square root of 144 is 12. And no negative sorry if we factor 2 here, then -1412 can be simplified to -76. Thus our answer is -76. If we look back at our choices, then option B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

Find the derivatives with respect to x of the following combinations at the given value of x.

h. √(f²(x) + g²(x)), x = 2

Key Concepts

Chain Rule

Power Rule

Sum Rule

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