Understanding Derivative: Solving for f'(x) in Different Functions

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In summary: The conversation discusses the power rule and its application in finding the derivative of a function. The power rule states that the derivative of a power function is the power multiplied by the coefficient and the base raised to the power minus one. The limit definition of a derivative can also be used to find the slope at a specific point, which is essentially what the power rule does.
  • #1
Paquita888
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Hi all

I am starting to learn Derivative,, does anybody wants to help me to show me

why does

f(x) = (x^2/ 3) - (3/x^2)
f'(x) = 2/3 x + 6/x^3 ?

and

f(x) = -3(2x^2 - 5x + 1)
f'(x) = 12x +15

I am really appreciate if anybody can show me how to get into the answer :) thanksss a lot!
 
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  • #2


You will have to show work before anybody starts contributing a solution. Try using the limit definition of a derivative for simple cases,

[tex]f'(a)=\lim_{h\to 0}{f(a+h)-f(a)\over h}[/tex]

Otherwise, you will need to know about the power rule.
 
  • #3


f(x) = -3(2x^2 - 5x + 1)
f'(x) = 12x +15

This is wrong.

f'(x) = -12x + 15 is correct.
 
  • #4


Here is the basic power rule, just apply it to each term: [tex] f(x)= n^x[/tex] therefore,
[tex] f'x,\frac{df(x)}{dx}= (n)x^{n-1}[/tex].

It's the same thing as: [tex]f'(a)=\lim_{h\to 0}{f(a+h)-f(a)\over h}[/tex]
 
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  • #5


Kevin_Axion said:
Here is the basic power rule, just apply it to each term: [tex] f(x)= n^x[/tex]
This is not the power rule. The function you show is an exponential function, not a power function.

A power function is a function of the form xn.

If f(x) = xn, then f'(x) = nxn-1
Kevin_Axion said:
therefore,
[tex] f'x,\frac{df(x)}{dx}= (x)n^{x-1}[/tex].

It's the same thing as: [tex]f'(a)=\lim_{h\to 0}{f(a+h)-f(a)\over h}[/tex]
No, it's not the same thing. You can use the definition of the derivative to get the formula for the derivative of a power function.
 
  • #6


Sorry I messed up the n and x, I fixed it. I know that [tex]f'(a)=\lim_{h\to 0}{f(a+h)-f(a)\over h}[/tex] is essentially the plugging in of points and finding the slope at that point. What I'm essentially saying is that the power rule gives a function in which you can find the slope at a point [tex]x[/tex] and that is what the limit formula does it finds the slope at a point, [tex]a[/tex]. The process is different but the end result is the same, [tex]f'x,\frac{df(x)}{dx}= (n)x^{n-1}[/tex] finds the slope at [tex]a[/tex], like the limit, if you plug in [tex]a[/tex]. Sorry for my mis-phrasing, I know what you mean though.
 
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  • #7


You still have nx in another part, if you want to fix that, too.
 

1. What is a derivative?

A derivative is a mathematical concept that represents the instantaneous rate of change of a function at a specific point. It is the slope of the tangent line at that point, and it measures how much the output of a function changes with respect to the input.

2. Why is it important to learn derivatives?

Derivatives are fundamental in many areas of mathematics and science, such as calculus, physics, economics, and engineering. They are also used extensively in real-world applications, such as optimization, modeling, and data analysis. Understanding derivatives allows for a deeper understanding of these fields and their applications.

3. How do I start learning derivatives?

To start learning derivatives, it is important to have a basic understanding of algebra, functions, and graphs. Then, you can begin by learning the definition of a derivative and its key properties, such as the power rule, product rule, and chain rule. Practice solving problems and working with different types of functions to gain a strong foundation.

4. What are some common mistakes to avoid when learning derivatives?

Some common mistakes to avoid when learning derivatives include using incorrect formulas or rules, not understanding the concept of limits, and not practicing enough problems. It is also important to pay attention to details and avoid careless errors.

5. How can I improve my understanding of derivatives?

To improve your understanding of derivatives, it is important to practice regularly, ask for help when needed, and try to explain the concepts to someone else. You can also explore different types of functions and their derivatives, and apply derivatives to real-world problems. Additionally, reviewing and revising your work can help identify areas for improvement.

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