Key to "What is the derivative of xex ?"

High School
Calculus and Analysis
Calculus 1
Derivatives

veksetz

Answered question

2021-12-19

What is the derivative of

x e^{x}

Answer & Explanation

Maria Lopez

Beginner2021-12-20Added 32 answers

To evaluate this derivative you use the product rule .
When you have a function that is the product of 2 functions f(x) and g(x) you can derive it as:

\frac{d}{d x} f (x) g (x) = f^{'} (x) g (x) + f (x) g^{'} (x)

In your case:

f (x) = x

and

f^{'} (x) = 1

g (x) = e^{x}

and

g^{'} (x) = e^{x}

Finally deriving the complete function we have:

\frac{d}{d x} x e^{x} = 1 e^{x} + x e^{x} = e^{x} (x + 1)

censoratojk

Beginner2021-12-21Added 46 answers

Using the Product Rule

[{(u v)}^{'} = u v^{'} + v u^{'}]

we can set

u = x

and

v = e^{x}

, and therefore,

u^{'} = 1

and

v^{'} = e^{x}

This makes the derivative

f^{'} (x) = x e^{x} + e^{x} = e^{x} (x + 1)

RizerMix

Expert2021-12-29Added 656 answers

Thanks! Good answer!

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