![]() Be aware of how the notation changes throughout the page. Be able to explain why the product of ∆r and ∆w is insignificant in this context. Review the formalization notes in the margin so you are able to clarify for students the meaning of all regions in questions 2 and 3. Students are provided a visual explanation of the product rule and are then asked to develop the product rule on their own. By itself, the product rule is generally not a challenge for calculus students, so we chose to make notational fluency, graphical representations, and connecting representations our focus today. The context for this lesson was interesting to our students and we had strong engagement in the lesson. Since we have two equations for h ’ ( x ) h’(x) h ’ ( x ), we can equate the two and solve for ’ ’ ’.Along with the derivative definitions and rules learned so far, the product rule is another foundational algorithm that students will use often throughout the AB and BC course. The inside term ’ ’ ’ represents the derivative of a product of functions. ![]() Notice that the second equation has the term 2 ’ 2’ 2 ’. Now we have two different equations for h ’ ( x ) h’(x) h ’ ( x ). Define F F F such that F ( x ) = f ( g ( x ) ) F(x) = f(g(x)) F ( x ) = f ( g ( x )) for every x x x, and let f f f and g g g be differentiable. We use the chain rule to differentiate compositions of functions. Through the first principle of derivatives, we’ve proved the product rule! So, if h ( x ) = f ( x ) ⋅ g ( x ) h(x) = f(x) \cdot g(x) h ( x ) = f ( x ) ⋅ g ( x ), where both f f f and g g g are differentiable functions, then the product rule is:ĭ d x ≠ d d x ⋅ d d x \frac h ’ ( x ) = Δ x → 0 lim f ( x + Δ x ) + Δ x → 0 lim Δ x g ( x + Δ x ) − g ( x ) + Δ x → 0 lim g ( x ) + Δ x → 0 lim Δ x f ( x + Δ x ) − f ( x ) The product rule derivative formula tells us that the derivative of a product of two differentiable functions is equal to the first function multiplied by the second function’s derivative, plus the second function multiplied by the first function’s derivative. If we can express a function in the form f ( x ) ⋅ g ( x ) f(x) \cdot g(x) f ( x ) ⋅ g ( x )-where f f f and g g g are both differentiable functions-then we can calculate its derivative using the product rule. How do you know when to use the product rule? The product rule allows us to differentiate two differentiable functions that are being multiplied together. Tim Chartier refers to the product rule as a game-changing derivative rule: ![]() The product rule allows us to calculate quickly the derivatives of products of functions that are not easily multiplied by hand-or that we can’t simplify any further.ĭr. ![]() The term “product of functions” refers to the multiplication of two or more functions. The product rule is a handy tool for differentiating a product of functions. Keep reading to learn how we use the product rule to simplify the differentiation process. The product rule is a useful addition to your mathematical toolbox. ![]()
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