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ryan_kidz

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G(x)= ((x^2+1)^4(x^5+3)^8(x+5)^9)/ ((x^4+3)^11+(x^4+2x+3)^5)

Question #1. Given a function G show that the logarithmic derivative of the product of two functions is the sum of the logaritmic derivatives of the two functions.

Question #2. Show that the logarithmic derivative of the quotient of the functions is the difference of their logarithmic derivatives.

Thanks for looking the question!
I really appreciate your helps!
 
0) What are you supposed to do with this function?

1) What is the "logarithmic derivative" of a function?

2) See (1).

Thank you.

Eliz.
 
Hello, ryan_kidz!

Like Eliz, I'm puzzled by the original function.
With a "+" in the denominator, log differentiation is not a pleasant prospect.

#1. Given a function \(\displaystyle y\) show that the logarithmic derivative of the product of two functions
is the sum of the logarithmic derivatives of the two functions.
Click to expand...
Is this supposed to be a simple demonstration?

And I assume that the logarithmic derivaitve of \(\displaystyle y\) is \(\displaystyle \frac{y'}{y}\) . . . by definition?


We have \(\displaystyle y\), a product of two functions, \(\displaystyle F\) and \(\displaystyle G:\;\;y\:=\:F\cdot G\)

Take logs: .\(\displaystyle \ln(y)\;=\;\ln(F\cdot G)\:=\:\ln(F)\,+\,\ln(G)\)

Differentiate implicitly: .\(\displaystyle \L\frac{y'}{y}\:=\:\frac{F'}{F}\,+\,\frac{G'}{G}\) . . . . is that it?
 
I already ad ded some addition on that problem to make it clearer.

I'm so about that and thnx
 
ryan_kidz said:
I already ad ded some addition on that problem to make it clearer.
Click to expand...
I'm sorry, but I can't see where you answered the questions I'd asked...?

Please reply with clarification. Thank you.

Eliz.
 
stapel said:
0) What are you supposed to do with this function?

1) What is the "logarithmic derivative" of a function?

2) See (1).

Thank you.

Eliz.
Click to expand...

I will call the quotient g' (x)/g((x) the logarithmic derivative of the function g

Thnx
 
I think I'd better rewrite the problem.
and i also add some more questions :)

f(x)= ((x^2+1)^4(x^5+3)^8(x+5)^9)/ ((x^4+3)^11+(x^4+2x+3)^5)

Question #1. Given a function g, we will call the quitient g'(x)/g(x) the logarithmic derivative of the function g. Show that the logarithmic derivative of the product of two functions is the sum of the logaritmic derivatives of the two functions.

Question #2. Show that the logarithmic derivative of the quotient of the functions is the difference of their logarithmic derivatives.

Question #3. Show that the logarithmic derivative of nth power of a function is n time the logarithmic derivative of the function.

Question #4. Use the idea of the above three parts to compute the ordinary (not the logarithmic) derivative of the function given at the beginning of this problem set. (Hint: compute the logarithmic derivative first)

Thanks for looking the question!
I really appreciate your helps!
 
I'm walking out the door in a minute, so I can't say much, but since you're just asking for how to get started....

1) Start by doing the things they refer to. They want you to show that g'(x)/g(x), where g is the sum of, say, f and h, is equal to f'(x)/f(x) + h'(x)/h(x). So write g(x) as the sum, and do the log derivative. For g(x) = f(x) + h(x), what is g'(x)?

2) Let g(x) = f(x)/h(x). Then what is g'(x)?

3) Let g(x) = f<sup>n</sup>(x). Then what is g'(x)?

4) Follow the instructions: start by finding the log derivative.

Already late: gotta go!

Eliz.
 
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