Product Rule

Definition

The product rule is a fundamental principle in calculus that provides a method for differentiating the product of two functions. Specifically, it states that if you have two differentiable functions, say $$f(x)$$ and $$g(x)$$, then the derivative of their product is given by $$ (f imes g)' = f' imes g + f imes g' $$, where $$f'$$ and $$g'$$ are the derivatives of $$f$$ and $$g$$, respectively. This rule is crucial when dealing with products in differentiation, allowing for efficient calculation of derivatives without needing to expand the product first.

5 Must Know Facts For Your Next Test

1. The product rule is essential for differentiating products of functions, making it easier to find derivatives without expanding complex expressions.
2. It can be applied to more than just two functions; for multiple functions multiplied together, the product rule can be extended using repeated application.
3. Both functions involved in the product rule must be differentiable at the point where the derivative is being calculated.
4. Understanding the product rule can significantly simplify problems in calculus where direct multiplication and differentiation would be cumbersome.
5. Common examples include finding the derivative of polynomial functions or trigonometric identities that are products of two different functions.

Review Questions

• How does the product rule facilitate the process of finding derivatives compared to expanding products first?
  • The product rule simplifies the process of finding derivatives by allowing us to calculate the derivative of a product directly without expanding it. This is especially useful for complex functions where expansion could lead to tedious calculations and potential errors. By applying the product rule, we can efficiently differentiate each function separately and combine their results, saving time and effort in our calculations.
• In what scenarios would you need to use both the product rule and chain rule together when differentiating a function?
  • You would need to use both the product rule and chain rule when differentiating a function that is a product of another function and a composite function. For example, if you have a function like $$h(x) = f(g(x)) imes g(x)$$, you'd apply the product rule to differentiate it as a whole while also applying the chain rule to find the derivative of $$f(g(x))$$. This combination allows for accurate differentiation of more complex expressions.
• Evaluate the expression $$h(x) = x^2 imes e^x$$ using the product rule and explain each step in detail.
  • To differentiate $$h(x) = x^2 imes e^x$$ using the product rule, we identify our functions: let $$f(x) = x^2$$ and $$g(x) = e^x$$. First, we find their derivatives: $$f'(x) = 2x$$ and $$g'(x) = e^x$$. Now applying the product rule, we get: $$h'(x) = f' imes g + f imes g' = (2x)(e^x) + (x^2)(e^x)$$. Combining these terms gives us: $$h'(x) = e^x(2x + x^2)$$. This process demonstrates how we can efficiently apply rules of differentiation to get our result clearly.

Related terms

Derivative:

A derivative represents the rate of change of a function concerning its variable, providing insight into how a function behaves at any point.

Chain Rule:

The chain rule is a formula used to compute the derivative of a composite function, stating that if a function is composed of other functions, its derivative can be found using the derivatives of the inner and outer functions.

Quotient Rule:

The quotient rule is a method for differentiating a quotient of two functions, stating that if you have two differentiable functions, then the derivative of their quotient is given by $$ (f/g)' = (f'g - fg') / g^2 $$.