Need Help With Derivative Function

In summary, a derivative function is a mathematical function that represents the rate of change of another function. It is important in fields such as science and engineering as it allows us to analyze the behavior of complex systems and understand how variables are changing. The derivative of a function can be found using calculus or online tools. A first derivative represents the rate of change of a function, while a second derivative represents the rate of change of the first derivative and can provide information about the concavity or curvature of a function. Derivative functions have many real-world applications in various fields, such as physics, engineering, economics, and statistics. They can be used to model and predict the behavior of systems, optimize processes, and solve problems involving rates of change.
  • #1
kwikness
17
0

Homework Statement


Find the Derivative Function of (4x - x[tex]^{2}[/tex])


The Attempt at a Solution



using formula:

[tex]
\frac{dy}{dx} = \frac{f(x + \Delta x) - f(x)}{\Delta x}
[/tex]



[tex]
\frac{4(x + \Delta x) - (-x^{2})}{\Delta x}
[/tex]

[tex]
\frac{4x + 4(\Delta x) + x^{2}}{\Delta x}
[/tex]

Not sure where to go from here..
 
Last edited:
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  • #2
Let d = [itex]\Delta[/itex]x.

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.
 
  • #3
First of all, you mean
[tex]
\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
[/tex]
as what you gave is just the difference quotient
[tex]
\frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}
[/tex]

Not sure how you got there in the first place. If I plug [itex]f(x) = 4x - x^2[/itex] into the formula you gave, I get
[tex] \frac{ [ 4(x + \Delta x) - (x + \Delta x)^2 ] - [4 x - x^2 ] }{ \Delta x }
= \frac{ 4 x + 4 \Delta x - x^2 - 2 x \Delta x - (\Delta x)^2 - 4 x + x^2 }{ \Delta x}
[/tex]
which has some terms you don't have. Now try again.
 
  • #4
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!
 
  • #5
HallsofIvy said:
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 :smile:
But I'm leaving him some work.
 
  • #6
Thanks, when I wrote it down I was missing a part of the equation. Gahhh! I always make stupid mistakes like that.
 
  • #7
You could just have used the Power Rule, but I guess you haven't learned it yet.
 

What is a derivative function?

A derivative function is a mathematical function that represents the rate of change of another function. It provides information about how a function is changing at a specific point.

Why do we need to use derivative functions?

Derivative functions are important in many fields of science and engineering because they allow us to analyze the behavior of complex systems. They help us understand how variables are related and how they are changing over time.

How do you find the derivative of a function?

The derivative of a function can be found using calculus. You can use the rules of differentiation, such as the power rule, product rule, and quotient rule, to find the derivative of a function. Alternatively, you can use computer software or online tools to calculate the derivative.

What is the difference between a first derivative and a second derivative?

A first derivative represents the rate of change of a function, while a second derivative represents the rate of change of the first derivative. In other words, the second derivative tells us how fast the rate of change is changing. It can provide information about the concavity or curvature of a function.

Can derivative functions be used in real-world applications?

Yes, derivative functions have many real-world applications in fields such as physics, engineering, economics, and statistics. They can be used to model and predict the behavior of systems, optimize processes, and solve problems involving rates of change.

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