What Is the Correct Way to Differentiate xe^2x?

[CrossFit415]

f(x) = xe^2x, x ε ℝ And determine the domain.

So I did...

f'(x) = xe^2x $\bullet$ d/dx 2

I applied the chain rule. I'm not sure if I did this right.

 

[iamalexalright]

you'll have to use the product rule AND the chain rule

 

[CrossFit415]

Ahh I see thank you

 

[Mark44]

you'll have to use the product rule AND the chain rule

And in that order.

 

[CrossFit415]

Okay cool!

 

[CrossFit415]

So I applied the product rule 1st;

f(x) = xe^2x
f'(x) = xe(2x) + xe(2)

Did I do this correctly?

 

[iamalexalright]

Not correct:

First, what is the derivative of e^(2x) ?

Second, if f and g are arbitrary functions of x, what is the derivative of f*g with respect to x (ie, what does the product rule say)?

 

[CrossFit415]

derivative of e^2x is 2e?

 

[iamalexalright]

Nope !

Here we have to use the chain rule but before we get there, what is the derivative of e^(x)?

 

[CrossFit415]

Just e^x

So...

e^2x = 2e^x ?

 

[iamalexalright]

Close but you are missing one thing.

Maybe if you saw another example it would become more clear...
What is the derivative of sin(2x)?

Or if you prefer by the definition of the chain rule:
$(f \circ g)' = f'(g) * g'$

In your case, what is f and what is g?
Then can you see your mistake?