How do you differentiate #f(x)=e^xe^(x^2)# using the product rule?

Question

Ezra Adams · Accepted Answer

You need more than the product rule ... add the chain rule too!

#f'=e^xe^(x^2)+e^xe^(x^2)(2x)=e^(x^2+x)(2x+1)# hope that helped

Axel Alvarez · Answer

undefined To differentiate $ f(x) = e^x e^{x^2} $ using the product rule:

1. Identify the functions $ u $ and $ v $: $ u(x) = e^x $ and $ v(x) = e^{x^2} $.
2. Apply the product rule: $ f'(x) = u'(x)v(x) + u(x)v'(x) $.
3. Find the derivatives of $ u(x) $ and $ v(x) $: $ u'(x) = e^x $ and $ v'(x) = 2xe^{x^2} $.
4. Substitute the derivatives into the product rule formula: $ f'(x) = e^x e^{x^2} + e^x \cdot 2xe^{x^2} $.
5. Simplify: $ f'(x) = e^x e^{x^2} + 2xe^{2x^2} $.

So, the derivative of $ f(x) = e^x e^{x^2} $ using the product rule is $ f'(x) = e^x e^{x^2} + 2xe^{2x^2} $.

Axel Alvarez · Answer

undefined To differentiate $ f(x) = e^x \cdot e^{x^2} $ using the product rule, you apply the formula:

$$ (uv)' = u'v + uv' $$

where $ u = e^x $ and $ v = e^{x^2} $. Then, you find the derivatives of $ u $ and $ v $ with respect to $ x $ and substitute them into the formula.

$$ u' = e^x $$
$$ v' = 2xe^{x^2} $$

Now, apply the product rule:

$$ (e^x \cdot e^{x^2})' = (e^x \cdot e^{x^2})' = (e^x)' \cdot e^{x^2} + e^x \cdot (e^{x^2})' $$

Substitute the derivatives:

$$ = e^x \cdot e^{x^2} + e^x \cdot 2xe^{x^2} $$

$$ = e^x \cdot e^{x^2} + 2xe^xe^{x^2} $$

Therefore, the derivative of $ f(x) = e^x \cdot e^{x^2} $ using the product rule is $ f'(x) = e^x \cdot e^{x^2} + 2xe^xe^{x^2} $.