differentiation

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April 13th 2010, 06:05 PM
tim_mannire

differentiation

I need help with the following derivatives

y= (e^1/x) / x^2

f(t) = t ln(5t)

thanks for your help
April 13th 2010, 06:17 PM
drumist

Could you show us what exactly is giving you trouble? What do you get when you attempt them?
April 13th 2010, 06:18 PM
harish21

Quote:

Originally Posted by tim_mannire

I need help with the following derivatives

y= (e^1/x) / x^2

f(t) = t ln(5t)

thanks for your help

USE QUOTIENT RULE ON THE FIRST ONE AND PRODUCT rULE ON THE SECOND ONE. SHOW YOUR WORK IF YOU GET STUCK!
April 14th 2010, 02:45 AM
tim_mannire

Using the rules you suggested i got

dy/dx = ((x^2)(e^1/x) - (e^1/x)(2x)) / x^4

f ' (t) = (1/5t) (t)

are they correct?
April 14th 2010, 03:50 AM
e^(i*pi)

Quote:

Originally Posted by tim_mannire

Using the rules you suggested i got

dy/dx = ((x^2)(e^1/x) - (e^1/x)(2x)) / x^4

f ' (t) = (1/5t) (t)

are they correct?

For dy/dx it appears you've not differentiated $e^{1/x}$ correctly

It's easier if you sub $u = \frac{1}{x}$ and then use the chain rule.

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For the second one remember the chain rule

$\frac{d}{dx} \ln [f(x)] = \frac{f'(x)}{f(x)}$
April 14th 2010, 05:25 AM
tim_mannire

thanks for your help. I have tried what you suggested but I just don't understand (Headbang). A complete working of the question using the rules would help me to understand.

thank you
April 14th 2010, 05:38 AM
e^(i*pi)

Quote:

Originally Posted by tim_mannire

thanks for your help. I have tried what you suggested but I just don't understand (Headbang). A complete working of the question using the rules would help me to understand.

thank you

$y = \frac{e^{1/x}}{x^2}$

For this we need to use the quotient rule which is: $y' = \frac{vu' - uv'}{v^2}$

In your case $u = e^{1/x}$ and $v = x^2$

I will differentiate the numerator as that's what seems to be the issue then you can try using my value of $u'$ in the quotient rule. FYI my final answer is $\frac{dy}{dx} = \frac{e^{1/x}(2x-1)}{x^4} = -\frac{e^{1/x}(1-2x)}{x^4}$

To find u' (du/dx) we need to work out $\frac{d}{dx} e^{1/x}$ .
For simplicity and clarity's sake let $t = \frac{1}{x}$ . From the chain rule we know that $\frac{du}{dx} = \frac{du}{dt} \cdot \frac{dt}{dx}$

Due to this substitution we can rewrite u as $u = e^t$ .

Differentiating this function is easy: $\frac{du}{dt} = e^t$

Similarly 1/x is also easy to differentiate: $\frac{dt}{dx} = -\frac{1}{x^2}$

From the chain rule above: $u' = \frac{du}{dx} = \frac{du}{dt} \cdot \frac{dt}{dx} = e^t \cdot \frac{1}{x^2}$

Since $t = \frac{1}{x}$ we get $u'$ in terms of x:

Therefore $u' = -\frac{1}{x^2}\,e^{1/x}$
April 14th 2010, 08:43 AM
drumist

Quote:

Originally Posted by tim_mannire

f(t) = t ln(5t)

For this one, it is

$f'(t) = t \cdot [\ln(5t)]' + [t]' \cdot \ln(5t)$

$= t \cdot \frac{5}{5t} + 1 \cdot \ln(5t)$