How to Differentiate Exponential Functions with Multiple Variables?

In summary, the conversation is discussing the differentiation of u = 2 (x-t2 / 3)et/3 and finding the values of du/dx and du/dt. The individual explains the use of the chain rule and product rule in finding these derivatives.
  • #1
andrey21
476
0
Differentiate the following:

u = 2 (x-t2 / 3)et/3

Here is my attempt

I have to find du/dx and du/dt

du/dx = 2 (x-t2 / 3)et/3

du/dt = -4t/9 (x-t2 / 3)et/3
 
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  • #2
It would help if you wrote u as 2xet/3 - (2/3)t2et/3. Since u is a function of both t and x, you will have to use the chain rule in your derivatives.

For example, if v = 3m2, then dv/dx = d/dx (3m2) = d/dm (3m2) (dm/dx) = 6m dm/dx.
 
  • #3
Ok so from what you have said:

u = 2xet/3 - 2/3 t2et/3

du/dx = 2/ e t/3

du/dt = 2/3 xe t/3 - 4/9 t e t/3
 
  • #4
Ok du/dx
There are errors in du/dt
 
  • #5
Ah yes I need to adopt the product rule on:

-2/3 t2 e t/3
 

FAQ: How to Differentiate Exponential Functions with Multiple Variables?

What is an exponential function?

An exponential function is a mathematical function in which the independent variable appears in the exponent. It is typically written as f(x) = ab^x, where a is the initial value and b is the growth factor.

How do you differentiate an exponential function?

To differentiate an exponential function, you can use the power rule, which states that d/dx (ax^n) = nax^(n-1). In the case of an exponential function, the derivative would be d/dx (ab^x) = a(ln b) * b^x.

What is the derivative of e^x?

The derivative of e^x is simply e^x. This is because e is a special number (approximately 2.71828) that has the property of being its own derivative.

How do you differentiate exponential functions with different bases?

To differentiate exponential functions with different bases, you can use the chain rule. For example, if you have an expression like f(x) = a^x, you can rewrite it as f(x) = (e^(ln a))^x and then use the chain rule to find the derivative as f'(x) = (ln a) * (e^(ln a))^x = (ln a) * a^x.

How can exponential differentiation be used in real life?

Exponential differentiation has many real-life applications, such as in finance, population growth, and radioactive decay. For example, it can be used to calculate compound interest, predict population growth, and determine how much of a radioactive substance remains after a certain amount of time.

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