Product rule example

Khan Academy

4 chapters5 takeaways9 key terms4 questions

Overview

This video explains and demonstrates the product rule for differentiation in calculus. It begins by stating the known derivatives of e^x and cos(x) and then introduces the general product rule: the derivative of a product of two functions u(x) and v(x) is u'(x)v(x) + u(x)v'(x). The video then applies this rule to find the derivative of e^x * cos(x), breaking down each component and simplifying the final expression. The goal is to make the abstract product rule more concrete through a practical example.

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Chapters

We need a method to find the derivative of a product of two functions, like e^x * cos(x).
We already know the derivatives of e^x (which is e^x) and cos(x) (which is -sin(x)).
The product rule provides a general formula for differentiating the product of two functions.

Understanding the product rule is essential because many functions in calculus are combinations of simpler functions, and this rule allows us to differentiate them.

The product rule states that the derivative of u(x) * v(x) with respect to x is u'(x)v(x) + u(x)v'(x).
This means you take the derivative of the first function and multiply it by the second function, then add the first function multiplied by the derivative of the second function.
The rule involves differentiating one part of the product at a time, ensuring both parts are accounted for in the final sum.

This formula provides a systematic way to break down the differentiation of a product into simpler, known derivatives, preventing common errors.

Derivative of u*v is (derivative of u)*v + u*(derivative of v).

Let u(x) = e^x and v(x) = cos(x).
The derivative of u(x) is u'(x) = e^x.
The derivative of v(x) is v'(x) = -sin(x).
Applying the product rule: (e^x)(cos(x)) + (e^x)(-sin(x)).

This step-by-step application makes the abstract product rule concrete, showing exactly how to substitute and combine known derivatives.

u=e^x, v=cos(x), u'=e^x, v'=-sin(x). Derivative = u'v + uv' = (e^x)(cos(x)) + (e^x)(-sin(x)).

The derivative expression is e^x * cos(x) - e^x * sin(x).
We can factor out the common term e^x.
The simplified derivative is e^x * (cos(x) - sin(x)).

Simplifying the derivative makes it easier to use in further calculations and provides a more elegant form of the answer.

Factoring e^x from e^x*cos(x) - e^x*sin(x) gives e^x(cos(x) - sin(x)).

Key takeaways

1The product rule is a fundamental calculus technique for differentiating functions that are products of two other functions.
2The derivative of a product u(x)v(x) is found by differentiating one function while keeping the other constant, and then adding the reverse: keeping the first constant while differentiating the second.
3Knowing the derivatives of basic functions (like e^x and trigonometric functions) is crucial for applying the product rule.
4The unique property of e^x (its derivative is itself) simplifies parts of the product rule calculation.
5Simplifying the final derivative expression, often by factoring, is an important step after applying the product rule.

Key terms

DerivativeProduct Rulee^xcos(x)sin(x)u(x)v(x)u'(x)v'(x)

Test your understanding

1What is the general formula for the product rule of differentiation?
2How does the product rule help in finding the derivative of a function like e^x * cos(x)?
3What are the derivatives of e^x and cos(x) that are needed for this example?
4Why is it useful to simplify the final derivative expression after applying the product rule?