Real Analysis, differentiation

In summary, the problem is asking to find g(0) and show that g'(x) = g'(0)g(x), given that g(x+y) = g(x)g(y) and g is differentiable. The solution involves taking the limit as h goes to 0 and factoring out g(x) to show that g'(x) is equal to the definition of g'(0).
  • #1
gaborfk
53
0
Solved: Real Analysis, differentiation

Homework Statement


If g is differentiable and g(x+y)=g(x)(g(y) find g(0) and show g'(x)=g'(0)g(x)

The Attempt at a Solution


I solved g(0)=1

and

I got as far as

[tex]
g'(x)=\lim_{\substack{x\rightarrow 0}}g(x) \frac{g(h)-1}{h}
[/tex]

but now I am stuck.

Thank you in advance
 
Last edited:
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  • #2
Never mind... Just did g'(0) and plug in...
 
  • #3
Your formula for g' is incorrect- though it may be a typo.
[tex]g'(x)= \lim_{\substack{h\rightarrow 0}}g(x)\frac{g(h)- 1}{h}[/tex]
where the limit is taken as h goes to 0, not x. Since "g(x)" does not depend on h, you can factor that out:
[tex]g'(x)= g(x)\left(\lim_{\substack{h\rightarrow 0}}\frac{g(h)-1}{h}\right)[/tex]
and you should be able to see that the limit is simply the definition of g'(0).
 

1. What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers and their properties. It involves the rigorous study of limits, sequences, series, continuity, differentiation, and integration of real-valued functions.

2. What is differentiation?

Differentiation is the mathematical process of finding the rate of change of a function with respect to its input variables. It involves calculating the derivative of a function, which represents the slope of a tangent line to the function's graph at a given point.

3. What are the basic rules of differentiation?

The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. These rules provide a systematic way to find the derivative of a wide range of functions.

4. How is differentiation used in real-world applications?

Differentiation has numerous real-world applications, including physics, engineering, economics, and statistics. It is used to model and analyze rates of change, such as velocity, acceleration, growth rates, and optimization problems.

5. What is the difference between differentiation and integration?

Differentiation and integration are inverse operations of each other. While differentiation calculates the rate of change of a function, integration calculates the area under the curve of a function. In simpler terms, differentiation is like taking a "zoomed in" view of a function, while integration is like taking a "zoomed out" view of a function.

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