# Differentiation of an exponential expression

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1. Nov 20, 2014

### Missy

1. The problem statement, all variables and given/known data

I need to differentiate the exponential function i = 12.5 (1-e^-t/CR) and I need to plot a table so that I can do a graph of i against t but i'm not sure how. (CR is the equivelant of Capacitance 20 Micro Fards and Resistance 300 Kilo Ohms)

2. Relevant equations

How do I differentiate this expression if it doesn'f follow the normal rules of differentiation (as follows)
1. y=ax^n becomes dy/dx = nax ^(n-1)
2. y = sinx becomes dy/dx = cosx
3. y = Ksinax becomes dy/dx = Kacosax
4. y = kcosax becomes kasinax
5. y = alnx becomes dy/dx = a/x

3. The attempt at a solution

i = 12.5 (1-e^-1/cr)
i = 12.5 (1-e^-1/300000*0.00002)
i = 12.5 (1-e^-0.167)
i = 12.5 (1-e^-0.167/1)
i = 12.5 e^-0.167
i = 2.09e^0.154

2. Nov 20, 2014

### SteamKing

Staff Emeritus
3. Nov 20, 2014

### Missy

Our teacher didn't cover any rules past the 5 that I listed, however, it has come up in the distinction work that I am doing. I will look over the link that you posted now.

4. Nov 20, 2014

### Staff: Mentor

No, this isn't right. You have left out t, the variable for time. For your table, find the current I for various values of t. For example, at t = 0.1 sec, you have
I= 12.5(1 - e^(-.1/(300000*0.00002)))
I get about .21 (A) for this.

The full problem description probably says what values to use for time.

5. Nov 20, 2014

### Missy

Hi Mark, Thanks for your assistance. I did have 1 as my time constant in my workings. I am struggling to reach the same answer as you for the 0.1. I really wish that my teacher had covered this for us and maybe I would see it more clearly.

6. Nov 20, 2014

### Staff: Mentor

Using R = 300,000 ohms and C = .00002 Farads, with t = .1 sec, you get
I = 12.5 (1 - e^(-.1/(300000 * 0.00002)))
= 12.5(1 - e^(-.1/6)) = 12.5(1 - .98347) = approx 0.207 (amp)

Do the same thing for whatever times are asked for in the problem. Or you could do the above calculation with t = 0.0, 0.1, 0.2, 0.3, and so on.

Part of the problem is in your calculations.

7. Nov 20, 2014

### Missy

I can see your answer, that's fantastic, thank you so so so much for helping me! I will plot the graph when I get in tomorrow and let you know how I got on :)

8. Nov 20, 2014

### Staff: Mentor

The five that you listed don't apply to your current equation. You need the rule for differentiating exponential functions as well as the chain rule. Both are listed in that wiki article.

9. Nov 20, 2014

### Staff: Mentor

If you manipulate your equation a little, you get:

$$ln(12-i)=ln(12)-\frac{t}{RC}$$

What is the slope of the straight line you get if you plot the data as ln(12 - i) versus t? What is derivative of ln(12-i) with respect to t?

Chet

10. Nov 21, 2014

### Missy

The slope of the straight line works out at a gradient of 0.65 so it's increasing. However, when I attempted the differentiation of the entire formula I got 3.89 which indicates that either my gradients wrong (unlikely) or my differentation is wrong (likely). Please can i have more help differentiating i = 12.5e(-7/6) I am struggling with the negative AND fractional power. Just to be clear the power is (-7) over (6) not minus (7/6)

11. Nov 21, 2014

### Missy

I know that the rule that is applying to my formula of i = 12.5 (1-e^(-t/cr) is:

"a(1-e^-bt)" which differentiates to abe^-bx

I have got as far as 12.5e^(-7/6) and I can see that I need to multiply 12.5 by 7 and then subtract 1 from the fraction but this gives me an overall answer that cant be right because the increase is too low, either that or my gradient is too high.

87.5e^-2.1666666667=10.02

12. Nov 21, 2014

### Staff: Mentor

You said you know how to differentiate the natural log. So what is dln(12-i)/dt in terms of di/dt and i?

Chet

13. Nov 21, 2014

### Staff: Mentor

Your derivative should be written in terms of t, not x. So d/dt(a(1 - e-bt) = abe-bt.
No, don't subtract 1. That's not part of the derivative.
Also, since the current and the derivative are both functions of time t, both will have different values for different t. In your calculation above, what value of t are you using?