Mathfolis

Product Rule

Unit 2 · Differentiation — Rules

The Product Rule

When differentiating a product of two functions, you cannot simply multiply their derivatives. The Product Rule says: derivative of first times second, plus first times derivative of second.

Product Rule

ddx[f(x)⋅g(x)]=f′(x)⋅g(x)+f(x)⋅g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)
Caution: Do NOT multiply derivatives: (fg)′ ≠ f′ · g′. This is a very common mistake.
Type 1

Basic Product Rule

Identify f and g, differentiate each, then combine using f′g + fg′.

Example 1
Find the derivative.
ddx[x3sin⁡x]\frac{d}{dx}[x^3 \sin x]
Example 2
Find the derivative.
ddx[x2ex]\frac{d}{dx}[x^2 e^x]

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Type 2

Evaluating the Derivative at a Point

Apply the product rule to find f′(x), then substitute the given x value.

Example 3
Find f′(1).
f(x)=(x2+1)(3x−2)f(x) = (x^2 + 1)(3x - 2)

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Type 3

Finding Where the Tangent Has Given Slope

Find f′(x) using the product rule, set it equal to the target slope, and solve for x.

Example 4
Find the derivative of f(x) = x ln x and determine the x > 0 where the tangent has slope 2.
f(x)=xln⁡xf(x) = x \ln x

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