Calculus / Unit 4: Differentiation Rules
4.2 The Product Rule
Notes
For a product of two functions: $\displaystyle \frac{d}{dx}[f g] = f'g + fg'$.
- The derivative of a product is NOT the product of the derivatives.
- Say it as "first' times second plus first times second'".
- Example: $\displaystyle \frac{d}{dx}[x^2(x + 3)] = 2x(x + 3) + x^2(1) = 3x^2 + 6x$. (Check by multiplying out first: $x^3 + 3x^2$.)
Practice
Which is the correct product rule?
The product rule adds two terms, each with one function differentiated.
Find $\displaystyle \frac{d}{dx}[x^3(2x - 1)]$.
$3x^2(2x - 1) + x^3(2) = 8x^3 - 3x^2$. The last choice is the original function, not its derivative.
Find the derivative of $(x^2 + 1)(x - 4)$ at $x = 1$.
The derivative is $2x(x - 4) + (x^2 + 1)(1)$. At $x = 1$: $2(-3) + 2 = -4$.
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