The product rule is a fundamental principle in calculus used to find the derivative of the product of two functions. It states that if you have two differentiable functions, say $$u(x)$$ and $$v(x)$$, then the derivative of their product $$u(x)v(x)$$ is given by the formula: $$u'v + uv'$$. This rule not only simplifies the differentiation process but also highlights the relationship between the functions being multiplied, making it essential for understanding how to differentiate more complex expressions.
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The product rule can be expressed as $$ (u imes v)' = u'v + uv' $$, where $$ u' $$ and $$ v' $$ are the derivatives of functions $$ u $$ and $$ v $$ respectively.
This rule applies to any two differentiable functions, making it a versatile tool in calculus when working with products.
If either function being multiplied has a zero derivative at a point, the overall product's derivative at that point will also be zero.
The product rule can be extended to products of more than two functions by applying it iteratively or using generalized forms.
It's crucial to remember that the product rule differs from simple multiplication; one cannot simply multiply derivatives together.
Review Questions
How does the product rule apply when differentiating a function that is expressed as a product of two other functions?
When differentiating a function that is expressed as a product of two other functions, you apply the product rule which states that the derivative of the product equals the derivative of the first function times the second function plus the first function times the derivative of the second. This means if you have functions $$ u(x) $$ and $$ v(x) $$, their derivative will be calculated using the formula: $$ u'v + uv' $$. This shows how both functions contribute to the overall rate of change.
Discuss how the product rule interacts with other rules of differentiation, such as the chain rule or quotient rule.
The product rule interacts closely with other differentiation rules like the chain rule and quotient rule by providing a systematic way to differentiate combined expressions. For instance, if you have a product where one or both functions are themselves composite functions, you can apply the chain rule to find their derivatives first and then use those results in the product rule. The quotient rule is similar; if you encounter division instead of multiplication, you'd switch to that method while knowing that each approach maintains consistency in calculus principles.
Evaluate a complex expression using both the product and chain rules together, demonstrating how they work in tandem.
To evaluate an expression like $$ f(x) = (x^2 + 1)( ext{sin}(x)) $$ using both the product and chain rules, first identify your functions: let $$ u(x) = x^2 + 1 $$ and $$ v(x) = ext{sin}(x) $$. Applying the product rule gives us: $$ f'(x) = u'v + uv' $$. Here, you'd find that $$ u' = 2x $$ and by applying the chain rule on $$ v(x) $$ you get that $$ v' = ext{cos}(x) $$. Thus, combining these results yields: $$ f'(x) = (2x)( ext{sin}(x)) + (x^2 + 1)( ext{cos}(x)) $$. This showcases how both rules are essential in navigating more complex derivatives.
Related terms
Derivative: The derivative represents the rate at which a function is changing at any given point and is a core concept in calculus.
The chain rule is another differentiation rule that allows for finding the derivative of composite functions, important for handling more complex expressions.