DerivativesEasy

Derivative — product rule

Calculate the derivative of f(x) = x² · eˣ using the product rule.

Given data

f(x) = x² · eˣ

Steps

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Identify the two factors: f(x) = u·v with u = x², v = eˣ.
Write your working — the numerical result must appear in the text (logica)
Calculate u' = (x²)' and v' = (eˣ)'.
Apply the product rule: f' = u'·v + u·v'.
Factor out eˣ. What is the final result?

Full worked solution

Identify the two factors: f(x) = u·v with u = x², v = eˣ.
$u = x^2,\quad v = e^x$
$f(x) = x^2 \cdot e^x$ . Choose $u = x^2$ and $v = e^x$ . The product rule states $(uv)' = u'v + uv'$ .
Calculate u' = (x²)' and v' = (eˣ)'.
$u' = 2x,\quad v' = e^x$
$u' = (x^2)'' = 2x$ (power rule), $v' = (e^x)'' = e^x$ (the exponential is its own derivative).
Apply the product rule: f' = u'·v + u·v'.
$f'(x) = 2x \cdot e^x + x^2 \cdot e^x$
$f'(x) = u'v + uv' = 2x\cdot e^x + x^2\cdot e^x$ . Applying the product rule with $u' = 2x$ and $v' = e^x$ .
Factor out eˣ. What is the final result?
$f'(x) = e^x (x^2 + 2x) = x e^x (x + 2)$
$f'(x) = e^x(2x + x^2) = e^x(x^2 + 2x) = \mathbf{x\,e^x(x + 2)}$ . Factoring out the common factor $e^x$ and then $x$ gives the simplified result.

Result:f'(x) = eˣ(x² + 2x) = x·eˣ(x + 2).

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