You have a function where and
I'm trying to get the derivative of 3xe^(2x)
I know the product rule and I know the derivative of e^2x is 2e^(2x).
So I was going to use the product rule, but I've only ever done that with two terms. Here, there are three.
thats a product rule and a chain rule
for the first. take derivative of 3x which is just 3 and keep second part. than. chain rule e^(2x) so e^ whatever is just e^ the whatever. so e^(2x) than multiply by 2 since the derivative of 2x is 2. keep the first part.
So you get:
3e^(2x) + 3xe^(2x)*2 which is 3e^(2x) + 6xe^(2x)
you can then simplify by pulling 3e^(2x) out of the answer
3e^(2x)*(1+2x) but that wouldnt really help unless your graphing and looking for critical numbers
Just in case a picture helps...
You have the chain rule - generally this pattern -
... wrapped inside the product rule...
... more visible here if we enclose some of the balloons in some more...
Hope this helps, or doesn't further confuse.
Don't integrate - balloontegrate!
Balloon Calculus: worked examples from past papers