Find the derivative function f' for the following functio...

Question

Find the derivative function f' for the following functions f.

f(x) = √3x+1; a=8

Accepted Answer

Welcome back, everyone. In this problem, given the function G of X equals the square root of 5 X + 2, find the derivative G at X equals 0. A says it's root 2 divided by 4, B 5 multiplied by root 2, D, 5 root 2 divided by 4, and D, 5 root 2 divided by 2. Now, to find the derivative G at X equals 0, we can use the limit definition, OK? Recall. That the derivative of G, to respect the X is going to be the limit as H approaches 0 of G X plus H. Minus G of X all divided by H. So what does that mean? Well, if we can figure out a value for a GFX plus H where we already know that GFX is the square root of 5 X + 2, we can plug it into the limit definition to solve for the derivative of G. Once we do that, then we can find X when, or sorry, we can find the value for our derivative at X equals 0. So first, let's try to find our derivative by by solving for a G X plus H. No, GX press H, OK. Is going to be equal. To the square root of 5 multiplied by x + h minus 2 because it plus 2 because it just means that wherever we see X we substitute X plus H and with that idea then this means that the derivative based on the definition forge of X. Is going to be the limit as H approaches 0 of the square root of 5 multiplied by X + H+ 2. OK. Minus the square root of 5 X plus 2. All divided by H. Now immediately if we were to substitute or directly substitute H for 0 in this expression, it would make the denominator 0 and thus our result will be undefined. We don't want that to happen. So let's see if we can rationalize our numerator to rearrange our expression so that not only H is in the denominator, that will probably help us to find the limit. So let's multiply it by our by the the the conjugate of our numerator here, which would be the square root of 5 multiplied by X + H+ 2 plus the square root of 5 X + 2, OK? And we're multiplying by that expression in the numerator and in the denominator. OK. Now, notice here that when we multiply in our numerator, OK. So right about the limit as H approaches 0, it's going to be equal to 5. Multiplied by X + H, OK, + 2 minus the expression 5 X + 2. And in our denominator, it's going to be H multiplied by the square root, OK, of 5 multiplied by X + H+ 2 minus. OK, the square root of 5, the square root, sorry, of 5 X plus 2. OK. So that's, that's the expression that we're going to get here, right? Now, let's go ahead and uh let's, let's let's try to simplify it some more. And now, in this case, OK. Not that. Uh, my apologies in our denominator, it should be plus, rather, it should be plus. OK. So now when we're going ahead here, when we're going to solve, that's going to be the limit as H approaches 0. In our numerator, notice that it's going to be If we expand our bracket, that's gonna be 5 X + 5 H + 2 minus 5 X minus 2. So 5 X and 2 are going to cancel, OK? And that is going to leave us with 5 H in our denominator divided by H multiplied by our expression, 5 multiplied by X + H+ 2 plus the square of 5 X + 2 in our denominator. And now we can cancel age from our numerator and denominator. To be left with the limit as H approaches 0, OK, of 5 divided by the square root of 5 multiplied by X plus H + 2. Plus the square root of 5 X + 2. Now, in this case, we can substitute H equals 0. We can directly substitute H in our expression. And when we do that here, notice that if H is 0, that's going to leave us with 5 divided by the square root of 5 X + 2. Plus the square root of 5 X + 2, and since they're adding, then this must be equal to 5 divided by 2 multiplied by the square root of 5 X plus 2. This is the derivative of GFX. Know that we have the derivative, then we can figure out the derivative at 0. OK. So at X equals 0, then G of 0 is going to be equal to 5 divided by 2 multiplied by the screw root of 5 multiplied by 0 + 2. OK. And that is gonna be equal to 5, OK, divided by 2. Multiplied by route 2 because 5 multiplied by 0 is 0 and know if we rationalize by multiplying by route 2 in our numerator and denominator. Route 2 cancels route 2, OK. And no, OK, well, sorry, let me, let me rewrite this here. Route 2 multiplied by root 2 is 2. OK. So that is gonna leave us with 5 root 2 divided by 4. So thus, that is the value of our derivative when X equals 0. If we look back at our answer choices, that tells us that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find the derivative function f' for the following functions f.
f(x) = √3x+1; a=8

Key Concepts

Derivative

Power Rule

Chain Rule

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