Having trouble differentiating exponential equations

In summary, an exponential equation is a mathematical expression with a variable in the exponent that represents a relationship between two quantities. The main difficulty in differentiating exponential equations is the variable in the exponent, requiring the use of the power and chain rules. To differentiate an exponential equation, you can rewrite it in logarithmic form and carefully label the different parts. Some common mistakes to avoid when differentiating exponential equations include forgetting to apply the chain rule, incorrectly applying the power rule, and not properly labeling the equation's parts. It is helpful to practice regularly and double-check your work to avoid these mistakes.
  • #1
102
0

Homework Statement


L(t)=15(0.5^(t/26))
Find the rate of L, when t=60


Homework Equations





The Attempt at a Solution



L'(t) = (15/26)(1/2)^(t/26)ln(1/2)
L'(60)=(15/26)(1/2)^(60/26)ln(1/2)

= -0.08

Did I do this right? If I did it wrong, please say where, I am having great trouble understanding this.
 
Physics news on Phys.org
  • #2
Yes you did it correctly. You say you have great trouble understanding "this". What is "this" exactly, how to differentiate an exponent?
 
  • #3
Cyosis said:
Yes you did it correctly. You say you have great trouble understanding "this". What is "this" exactly, how to differentiate an exponent?

I guess I was just lucky to be honest, since I did the question with a friend. My problem is applying the chain rule, and knowing where/when to put ln.

Here's another, that I have not finished yet (don't know how)

Flow of lava from a volcano is modeled by

l(t)= 12(2-0.8^t)

l is dist from crater in km
t is time in hours

how fast is the lava traveling down hillside after 4hours,

well, firstly i need to differentiate l(t), then solve for t

Can somebody give me hints for what to do with the t? I'm not quite sure..
 
  • #4
0.8^t=e^(ln(0.8)*t). The derivative of e^(ct) with respect to t is what? Use the chain rule. Powers of constants are just exponentials.
 
  • #5
Use this derivative shortcut for exponential functions:
dy/dx a^u=ln(a)*a^u*du //*du is the derivative of u; this is where you use the chain rule
For a=e, dy/dx e^u=e^u*du

You are supposed to find the rate at which lava flows down the hill at 4 hours. Since t is the time in hours and l(t) is the distance of the lava from the crater in km, you need
to find l'(4) to get the rate. It is not necessary to "solve" for t: t=4. If you were told the rate and were supposed to find the time, t, you would use algebra to solve for t, much as any other equation. Since l'(t) is an exponential function, you would have to use logarithms and the property that log(a^b)=b log(a), or that log(a)/log(b)=log_b(a).
 
  • #6
An exponential is about the easiest function to differentiate!

(ex)'= ex and (ax)'= ax ln(a).

More generally, by the chain rule, (af(x))'= af(x)ln(a) f'(x).
 
  • #7
l(t)= 12(2-0.8^t)

so l'(t) = 12(2-0.8^(t) ln 0.8)

If that's true, it doesn't seem to get me anywhere.
 
  • #8
Draggu said:
l(t)= 12(2-0.8^t)

so l'(t) = 12(2-0.8^(t) ln 0.8)

If that's true, it doesn't seem to get me anywhere.
That's not quite right, why is there still a 2 there?
 
  • #9
Hootenanny said:
That's not quite right, why is there still a 2 there?

Ah yes.

so l'(t) = 12[-0.8^(t)] ln 0.8)

..
..

So when I sub in 4 for t, I get 1.1. Is this correct?
 
  • #10
Draggu said:
Ah yes.

so l'(t) = 12[-0.8^(t)] ln 0.8)

..
..

So when I sub in 4 for t, I get 1.1. Is this correct?
Indeed you do.
 
  • #11
Because the question asked
"how fast is the lava traveling down hillside after 4hours"

One more question, just needs clarifying.

For example if I had (1/2)^(t/138)

The derivative would be 1/138(1/2)^t/138 ln 1/2 ?

I'm having trouble differentiating an exponent with a denominator
 
  • #12
Draggu said:
Because the question asked
"how fast is the lava traveling down hillside after 4hours"
Apologies, I was looking at your OP.
 
  • #13
Draggu said:
Because the question asked
"how fast is the lava traveling down hillside after 4hours"

One more question, just needs clarifying.

For example if I had (1/2)^(t/138)

The derivative would be 1/138(1/2)^t/138 ln 1/2 ?

I'm having trouble differentiating an exponent with a denominator

That derivative would be correct. With a denominator, remember the chain rule" dy/dx=dy/dt*dt/dx.

To put it simply, you multiply the derivative of the function (the a^x*ln(x) ) by the derivative of the exponent.

So for a derivative with a denominator, you would have a^(x/b)*ln(a)*dx/b.

Examples: dy/dx 2^(x/20)=2^(x/20)*ln(2)*1/20
dy/dx 3.4^[(x^2)/85]=3.4^[(x^2)/85]*ln(3.4)*(2x)/85
dy/dx 3-5^[(x^4)/220]=-5^[(x^4)/220]*ln(5)*[(x^3)/55]

Does that help?
 

1. What is an exponential equation?

An exponential equation is a mathematical expression that includes a variable in the exponent, such as y = ab^x. It represents a relationship between two quantities where one quantity is increasing or decreasing at a constant rate.

2. What makes differentiating exponential equations difficult?

The main challenge in differentiating exponential equations is that the variable is in the exponent, making it difficult to apply traditional differentiation rules. Additionally, there are multiple forms of exponential equations, such as exponential growth and decay, which require different approaches to differentiation.

3. How can I differentiate an exponential equation?

To differentiate an exponential equation, you can use the power rule, which states that the derivative of x^n is nx^(n-1). In the case of exponential equations, you would also need to apply the chain rule to account for the variable in the exponent. It is important to identify which type of exponential equation you are dealing with and use the appropriate differentiation method.

4. Are there any tips for differentiating exponential equations?

One helpful tip is to rewrite the exponential equation in logarithmic form, as this can make it easier to apply differentiation rules. It is also important to carefully identify and label the different parts of the equation, such as the base and the exponent, to avoid mistakes in the differentiation process. Practice and familiarizing yourself with different types of exponential equations can also improve your differentiation skills.

5. What are some common mistakes to avoid when differentiating exponential equations?

Some common mistakes when differentiating exponential equations include forgetting to apply the chain rule, incorrectly applying the power rule, and forgetting to label the different parts of the equation. It is also important to be careful with negative exponents and to properly handle constants when differentiating. It is always helpful to double-check your work and practice regularly to avoid these mistakes.

Suggested for: Having trouble differentiating exponential equations

Back
Top