How Do You Solve the Derivative of \(4x - x^2\)?

[kwikness]

Homework Statement

Find the Derivative Function of (4x - x^{2})

The Attempt at a Solution

using formula:

<br /> \frac{dy}{dx} = \frac{f(x + \Delta x) - f(x)}{\Delta x}<br />



<br /> \frac{4(x + \Delta x) - (-x^{2})}{\Delta x}<br />

<br /> \frac{4x + 4(\Delta x) + x^{2}}{\Delta x}<br />

Not sure where to go from here..

 

[EnumaElish]

Let d = \Deltax.

f(x+d) - f(x) = 4(x+d)-(x+d)^2 - 4x + x^2

which simplifies to (4 - d - 2x)d (you should derive this). The rest should be easy.

 

[CompuChip]

First of all, you mean
<br /> \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}<br />
as what you gave is just the difference quotient
<br /> \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}<br />

Not sure how you got there in the first place. If I plug f(x) = 4x - x^2 into the formula you gave, I get
\frac{ [ 4(x + \Delta x) - (x + \Delta x)^2 ] - [4 x - x^2 ] }{ \Delta x }<br /> = \frac{ 4 x + 4 \Delta x - x^2 - 2 x \Delta x - (\Delta x)^2 - 4 x + x^2 }{ \Delta x}<br />
which has some terms you don't have. Now try again.

 

[HallsofIvy]

That's because you haven't finished the algebra! You have 4x and -4x in the numerator! You have -x² and x² in the numerator!

 

[CompuChip]

That's because you haven't finished the algebra! You have 4x and -4x in the numerator! You have -x² and x² in the numerator!

Exactly, and the -4x and -x² just happen to be some of the terms kwikness is missing
But I'm leaving him some work.

 

[kwikness]

Thanks, when I wrote it down I was missing a part of the equation. Gahhh! I always make stupid mistakes like that.

 

[Hydrargyrum]

You could just have used the Power Rule, but I guess you haven't learned it yet.