Use the definition of the derivative to determine d/dx(ax²+bx+...

Question

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

Accepted Answer

Welcome back, everyone. Determine the derivative of the function G of X equals KX2 + L X + M using the definition of the derivative where K L and M are constants, and we're given four answer choices A says 2K plus L, BKX plus L, C2KX plus L and D 2K plus M. So let's write the definition of the derivative for this specific function. If we want to identify the derivative in the form of G X, we have to take the limit as H approaches 0 of the different quotient, and that's g at point X plus H minus G of X, and we are dividing that by H. All that we have to do is identify G of X plus H, and we can do that by replacing every X with X + H. So we're going to get K multiplied by X + H2 + L X. So that's L multiplied by X plus H plus M. And now what we're going to do is simplify this expression. So K multiplied by x2 + 2 H X plus H2. We can get that by squaring the sum of two terms and then we're going to distribute. So we're going to get plus LX plus HL and finally we have M, right? And then we want to rewrite G of X, that's KX2 + L X + M, and then we want to find the difference between those, right? So we're going to get G of X plus H minus G of X. It's going to be equal to, let's distribute the terms that's KX2 + 2. K H X plus k H2 + L X plus H L plus M and we are subtracting G of X. So we're going to add a negative sign for each term minus KX2 minus LX minus M. And now let's simplify. We have a lot of terms, but we can notice that first of all, we have KX2 minus KX squad. We have M minus M, and we also have positive LX and negative LX. So we have 2 KHX plus KH2 plus H L. And we're going to substitute this into the numerator. So we're going to get limit as H approaches 0 of 2 KHX plus KH2 + HL. All of that divided by H. and we can now understand that it would be better to factor out the H, right? So we're going to factor out the H and we're going to end up with 2 KX plus. KH plus L. And we can see that we can actually cancel out the age and attempt the direct substitution, right? So if we attempt the direct substitution from here, we're going to get 2 K X plus K multiplied by 0 because H approaches 0, so 0K simply gives us 0, right? And then we have plus L. And this gives us 2 K X plus L as our final answer. So considering the final answer, we can conclude that the correct answer choice to this problem corresponds to the answer choice C. Thank you for watching.

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

Key Concepts

Definition of the Derivative

Power Rule

Linearity of Derivatives

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