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f(x) = x² · eˣ
Step 1: Apply product rule
d/dx[x² · eˣ] = x² · d/dx[eˣ] + eˣ · d/dx[x²]
Step 2: Simplify
= x² · eˣ + eˣ · 2x = eˣ(x² + 2x)
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Problem
Differentiate f(x) = x² · eˣ
Step 1: Identify the rule
Use the product rule: (uv)' = u'v + uv'
Step 2: Apply the product rule
d/dx[x² · eˣ] = (d/dx[x²]) · eˣ + x² · (d/dx[eˣ])
Step 3: Differentiate each part
= 2x · eˣ + x² · eˣ
Step 4: Factor (optional)
= eˣ(2x + x²)

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Why do we use the product rule here?
Great question! We use the product rule because f(x) = x² · eˣ is the product of two separate functions: x² and eˣ. When you multiply functions, their derivatives don't just multiply — you need the product rule: (uv)' = u'v + uv'.

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