# [Finite Maths] Derivative Problems

• Nov 29th 2009, 01:45 AM
Shamanistic
[Finite Maths] Derivative Problems
Hey guys. I'm a newbie here and if this thread is in the wrong place please forgive me. I didn't know where to post derivative questions, so I figured this place would be the most logical.
I was going through the review questions I had to complete, and off all the 25 examples, Y'm seriously stuck on a few.
I'm supposed to find the derivatives of the following questions:

1- f(x) = (e^x + e^-x) / (e^x - e^-x )

2- g(x) = e^3x-1 . e^x-2 . e^x

3- f(x) = 1 / x ln x

Solving any of them is still a big help, and I'd appreciate it if you could show your workings so I could at least understand what you did and why.

Thanks a lot :)
• Nov 29th 2009, 02:35 AM
robeuler
Quote:

Originally Posted by Shamanistic
Hey guys. I'm a newbie here and if this thread is in the wrong place please forgive me. I didn't know where to post derivative questions, so I figured this place would be the most logical.
I was going through the review questions I had to complete, and off all the 25 examples, Y'm seriously stuck on a few.
I'm supposed to find the derivatives of the following questions:

1- f(x) = (e^x + e^-x) / (e^x - e^-x )

2- g(x) = e^3x-1 . e^x-2 . e^x

3- f(x) = 1 / x ln x

Solving any of them is still a big help, and I'd appreciate it if you could show your workings so I could at least understand what you did and why.

Thanks a lot :)

Do you know the quotient rule?

It says that if a(x) and b(x) are differentiable functions, the derivative of
a(x)/b(x) is (a'(x)b(x)-a(x)b'(x))/(b(x)^2) where a'(x) is the derivative of a(x).

Apply this for number 1 with a(x)=e^x + e^-x and b(x)=e^x - e^-x

a'(x)=e^x - e^-x
b'(x)=e^x + e^-x

so f'(x)= ((e^x - e^-x)^2 -(e^x + e^-x)^2)/((e^x - e^-x)^2)

for number 2)-is that a product?
If so use the product rule: if a(x) and b(x) are differentiable functions, the derivative of a(x)b(x)=a'(x)b(x) + a(x)b'(x).

For number 3) use the quotient rule again with a(x)=1 (remember that a'(x)=0 for constants!) and b(x)=x(ln(x)). An extra hint: you will need the product rule for b'(x).
• Nov 29th 2009, 05:06 AM
Shamanistic
thanks a lot for your help!

For your solution for problem 2, I have a question.

2- g(x) = e^3x-1 . e^x-2 . e^x

You said I should use the product rule here. What do I do when there are 3 parts in the equation?

Is it something like this:
f'(x) = a'(x)b(x)c(x) + a(x)b'(x)c(x) + a(x)b(x)c'(x) ?

Thanks again :)
• Nov 29th 2009, 05:07 AM
Raoh
i'll take the $3^{rd}$ question
we have $f(x)=\frac{1}{x \ln x}=\left ( x\ln x \right )^{-1}$
and u should know that $\left (f^{r} \right )'=rf^{r-1}f^{'}$,
so $\forall x\in (0,\infty)$ , $\left ((x\ln x)^{-1} \right )'=$ $-1(x\ln x)^{-2} (\ln x+1)$ $=-\frac{1}{\left (x\ln x \right )^{2}}\times\left (\ln x+1 \right )$ $=-\frac{(\ln x+1)}{(x\ln x)^2}$
• Nov 29th 2009, 05:26 AM
Raoh
Quote:

Originally Posted by Shamanistic
thanks a lot for your help!

For your solution for problem 2, I have a question.

2- g(x) = e^3x-1 . e^x-2 . e^x

You said I should use the product rule here. What do I do when there are 3 parts in the equation?

Is it something like this:
f'(x) = a'(x)b(x)c(x) + a(x)b'(x)c(x) + a(x)b(x)c'(x) ?

Thanks again :)

if u mean $g(x)=\exp(3x-1)\exp(x-2)\exp x$
then u have also $g(x)=\exp(5x-3)$
• Nov 29th 2009, 06:49 AM
Shamanistic
Seriously I've got to learn to look at these kinda problems the way you guys do. It's so simple :)

Thanks a dozen!
• Nov 29th 2009, 10:06 AM
Shamanistic
Can anybody help me find the derivative of this problem as well?

K(x) = (x^2 / 1+x)^2

Should the square be applied inside the brackets, or should I solve it like this:

K'(x) = 2.(x^2 / 1+x) . (2x/1)