1. ## Differenntiate equation, help

Hello, Can anyone please help me to differentiate $\displaystyle e^{(10.234 - 6.984*0.96^{(t+3.3)})}* 1000 * 0.96^t$

Any help would be appreciated.

Thanks

2. Originally Posted by dgamma1
Hello, Can anyone please help me to differentiate $\displaystyle e^{(10.234 - 6.984*0.96^{(t+3.3)})}* 1000 * 0.96^t$

Any help would be appreciated.

Thanks
Let's make it look a little nicer by bringing all the constants out front.

$\displaystyle 1000 e^{10.234} [.96^t][e^{(-6.984).96^{t+3.3}} ]$

Leave the constants for now

$\displaystyle [.96^t][e^{(-6.984).96^{t+3.3}} ]$

$\displaystyle [.96^t][e^{(-6.984)(.96^{3.3}).96^{t}} ]$

Let $\displaystyle y = [.96^t][e^{(-6.984)(.96^{3.3}).96^{t}} ]$

$\displaystyle lny = ln [.96^t] + ln[e^{(-6.984)(.96^{3.3}).96^{t}} ]$

$\displaystyle lny = t ln(.96) + (-6.984)(.96^{3.3}).96^{t} ln(e)$

$\displaystyle \frac{1}{y} y^{ \prime} = ln(.96) - (6.984)(.96^{3.3})[.96^{t}]^{\prime}$

To find the unknown derivative on the right hand side

Let $\displaystyle p = .96^t$

$\displaystyle lnp = t ln .96$

$\displaystyle \frac{1}{p} p^{ \prime } = ln .96$

$\displaystyle p^{ \prime } = .96^t ln .96$

Subbing back in

$\displaystyle \frac{1}{y} y^{ \prime} = ln(.96) - (6.984)(.96^{3.3})[.96^t ln .96 ]$

Finally,

$\displaystyle y^{ \prime} = [.96^t][e^{(-6.984)(.96^{3.3}).96^{t}} ] ln(.96) - (6.984)(.96^{3.3})[.96^t ln .96 ]$

But remember we have that constant out front, so in total we get

$\displaystyle y^{ \prime} = 1000 e^{10.234} [.96^t][e^{(-6.984)(.96^{3.3}).96^{t}} ] ln(.96) - (6.984)(.96^{3.3})[.96^t ln .96 ]$

3. wow thanks for that. although there is a step I don't understand:

1. going from here:

$\displaystyle lny = t ln(.96) + (-6.984)(.96^{3.3}).96^{t} ln(e)$

to here: what did you do with the t in front of the Ln, and where did the Ln go from the left side.

$\displaystyle \frac{1}{y} y^{ \prime} = ln(.96) - (6.984)(.96^{3.3})[.96^{t}]^{\prime}$

Thanks for all your help so far!

4. Originally Posted by dgamma1
wow thanks for that. although there is a step I understand:

. going from here:

$\displaystyle lny = t ln(.96) + (-6.984)(.96^{3.3}).96^{t} ln(e)$$\displaystyle to here: what did you do with the t in front of the Ln, and where did the Ln go from the left side. \displaystyle \frac{1}{y} y^{ \prime} = ln(.96) -$$\displaystyle (6.984)(.96^{3.3})[.96^{t}]^{\prime}$

Thanks again for all your help!

No problem mate.

This is known as implicit differentiation.

Let's say we have

$\displaystyle y = lnt$

The derivative of $\displaystyle lnt$ with respect to t is $\displaystyle \frac{1}{t} t` = \frac{1}{t} (1)$

In this case, we have

$\displaystyle lny$ and the derivative of which (with respect to t) is $\displaystyle \frac{1}{y} y^{ \prime }$

In this case, however, y depends on t so $\displaystyle y^{ \prime }$ doesn't equate to 1 like in the above example. So we must keep it.

By the same token,

$\displaystyle tln(.96)$

We are taking the derivative of the above with respect to t, which is equal to

$\displaystyle ln(.96)$

So basically we're taking the derivative of the left side with respect to t, and of the right side with respect to t. But since the left side is depdent on t and we dont have an equivilent expression for it, we leave it as $\displaystyle y^{ \prime }$