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

Question

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

f(x) =3x²+2x−10; a=1

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the derivative of the function K at the given value D, where K X equals 2 X2 minus 4 X + 6 and the D equals 0. A says it's a 0, B4, C-4, and the D says it's 8. Now to calculate the derivative of the function K, then we can use the definition of the limit. Recall. That the derivative of our function in this case the derivative of K X is going to be equal to the limit as H approaches 0 of K X plus H minus K X all divided by H. So what this means is that we'll need to find K X plus H. We already know KX and substitute both of those into our limit definition to help us find our derivative. Once we find the derivative, then we can substituted into the derivative to help us figure out what it is at that given valued equals 0. So first, let's find KFH plus X, OK? We know that K X equals 2 X2 minus 4 X + 6. So now everywhere we see X, we're going to substitute X plus H. It's going to leave us with this expression, OK. And now if we go ahead and simplify, OK, then if we expand X + H squared, it's gonna be X2 + 2 XH plus H squared. If we expand -4 across X + H, that's -4 X minus 4H, and then we have + 6. And if we distribute 2 across our squared terms, that's going to be 2 X2 + 4 XH plus 2 H squad. OK, minus 4 X minus 4 H plus 6. Now that we have an expression for a K X +8, let's substitute it into our derivative. Now, this tells us that the derivative then of K of X is going to be the limit as it approaches 0, OK. Of K X plus H, that's 2 X2 + 4 XH plus 282 minus 4 X minus 4 H + 6. OK, minus K X minus 2 X2 minus 4 X + 6. OK. And now all of this. Is being divided by age, OK. Now, let's expand and try to simplify. Here we can remove our brackets, OK? And if we distribute the negative term across K of X, it's going to be minus 2 X2 + 4 X, OK, minus 6. Now when we look at our numerator, 2 X2 minus 2 x 2 is 0, negative 4 X plus 4 X is 0, and 6 minus 6 is 0. So now when we get rid of those terms, then we're left with the limit as H approaches 0 of 4 XH, OK, plus 2 H2, OK, minus 4 H and that's all being divided by H. But now we can divide through our entire term by H because they all have H in them. So if we factor out H in the numerator, so that's gonna be H multiplied by 4 X plus 2H minus 4, OK. All divided by H. and when we cancel the H term, then we're left with the limit as it approaches 0 of 4 X plus 2H minus 4. Now what is that going to be? Well, if we take the limit as H approach is 0 by substitutingh, that's 4 X plus 2 multiplied by 0 minus 4, which is going to leave us with a derivative of K X being equal to 4 X minus 4. Now that we have the derivative of K of X, we can figure it out when D equals 0. And at D equals 0, OK, then. The sorry, K D is going to be equal to K 0, and if we substitute 0 into K or derivative, that's 4 multiplied by 0 minus 4, which equals -4. Thus, the derivative of the function K at the given value D equals 0 is going to be -4. If we look back at our answer choices, that means 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) =3x²+2x−10; a=1

Key Concepts

Derivative

Power Rule

Evaluating Derivatives at a Point

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