How do I find the derivative of a function with a given limit and function rule?

In summary, the conversation discusses finding the value of f(0) and finding the derivative of the function f(x), given that f(x+y)=f(x)+f(y)-2xy and the limit f(x)/h=7 as h approaches 0. The first part (a) is already solved and f(0) is determined to be 0. The second part (b) requires using the definition of the derivative to find the derivative of f(x) and the third part (c) involves taking the integral of the derivative to find the function f(x). The conversation also mentions confusion with understanding how to do limits.
  • #1
feuerwasser
4
0

Homework Statement



for all numbers x and y, let f be a function such that f(x+y)=f(x)+f(y)-2xy and such that the limit f(x)/h=7
h[tex]\rightarrow[/tex]0
a. find f(0). Justify your answer.


b. Use the definition of the derivative to find the derivative of f(x)

c. Find f(x)

i already did section a and and got f(0) = 0, which my teacher said was correct. and i know that to get section c i would just take the integral of b. but i have absolutely no clue how to get b. would i just take the derivative of f(x+y)=f(x)+f(y)-2xy and set f(x+y) to 0? or do i take the derivative of
 
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  • #2
The problem specifically says "use the definition of the derivative" which is, of course,
[tex]\lim_{h\rightarrow 0} \frac{f(x+ h)- f(x)}{h}[/tex]
You are told that f(x+y)= f(x)+ f(y)- 2xy so f(x+h)= f(x)+ f(h)- 2xh.
[tex]\frac{f(x+h)- f(x)}{h}= \frac{f(h)- 2h}{h}= \frac{f(h)}{h}- 2[tex]
What is the limit of that?
 
  • #3
would the limit be as h approaches zero, since that would cause it to be undefined? my teacher really didn't explain how to do limits so I'm kinda lost
 

What is a derivative?

A derivative is a mathematical concept that represents the slope or rate of change of a function at a specific point. It is calculated by finding the limit of the ratio of change in the output of the function to the change in the input of the function as the change in input approaches zero.

What is the difference between a derivative and a limit?

A derivative is a specific type of limit that represents the instantaneous rate of change of a function at a specific point. A limit, on the other hand, represents the value that a function approaches as its input approaches a certain value.

Why are derivatives important?

Derivatives are important in many fields of science, including physics, economics, and engineering. They allow us to model and understand the rate of change of real-world phenomena. They are also essential for optimization problems, such as finding the maximum or minimum value of a function.

How do you find the derivative of a function?

The derivative of a function can be found using the rules of differentiation, which involve taking the limit of a ratio. These rules include the power rule, product rule, quotient rule, and chain rule. In some cases, it may also be necessary to use implicit differentiation or logarithmic differentiation.

What are some applications of derivatives?

Derivatives have numerous applications in science, technology, and everyday life. They are used to model and understand the behavior of physical systems, such as the motion of objects, the growth of populations, and the spread of diseases. They are also used in economics to analyze the behavior of markets and in engineering to optimize designs and control systems.

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