Elihu5991 said:Homework Statement
Find the derivative of these e^x functions [paraphrased]
Homework Equations
[itex]x^{2} e^{x}[/itex]
The Attempt at a Solution
[itex]2x e^{x}[/itex]
This is how I believe it to be correct from my understanding. It does feel wrong, with the answers agreeing with the feeling.
Elihu5991 said:Can't believe I didn't notice the use of the product rule. As I was progressing through the question, I wasn't sure of one new thing: is two [itex]e^{x}[/itex] equal to [itex]e^{x}[/itex] or [itex]2e^{x}[/itex]? I reckon it's the latter, but the answers are in the simplest form and doesn't seem to fit; unless I've made another minute error.
mtayab1994 said:Well try to look at x^2e^x as u=x^2 and v=e^x
so (uv)'=u'v+v'u. Now just compute that and you will find your derivative.
You should get 2xe^x+x^2e^x=x(2e^x+xe^x)=xe^x(x+2)
Elihu5991 said:Did you differently simplify it? The answer in my book: [itex]xe^{x}(2+x)[/itex]
Elihu5991 said:Thankyou everyone for your help.
P.S How do I mark this topic as [SOLVED] ?
The derivative of e^x is simply e^x. This means that the derivative of a function with e^x as its base will always be e^x, regardless of the power or exponent.
To find the derivative of a function with e^x as its base, you simply use the chain rule. This means that you multiply the function by the derivative of the exponent, which is always 1, and then multiply by the derivative of the entire function, which is e^x.
Sure! Let's say we have the function f(x) = 2e^x. Using the chain rule, the derivative would be f'(x) = 2 * 1 * e^x = 2e^x. As you can see, the derivative is simply e^x, just like we discussed in the first question.
E^x is important in calculus because it is its own derivative, meaning that its derivative is itself. This makes it a useful tool for solving more complex functions, as well as for applications in physics and engineering.
Yes, there are a few other important properties and rules for derivatives of e^x. For example, the derivative of a constant times e^x is simply that constant times e^x. Additionally, the derivative of ln(e^x) is simply 1. These properties can be helpful when solving more complex functions involving e^x.