Product Rule of x = r cos()

In summary, the conversation discusses the use of the product rule and chain rule in finding the derivative of cos(theta) with respect to t. The chain rule is used because theta is a function of t, resulting in the inclusion of d\theta/dt in the final derivative.
  • #1
TheDoorsOfMe
46
0

Homework Statement



r = r(t)
[tex]\theta[/tex] = [tex]\theta[/tex](t)

x = r cos([tex]\theta[/tex])

dx/dt =dr/dt cos([tex]\theta[/tex]) - r sin([tex]\theta[/tex]) d[tex]\theta[/tex]/dt

The Attempt at a Solution



Where does the d[tex]\theta[/tex]/dt come from at the end of the derivative? I know I'm using product rule here because r and theta are both functions of t. But, the derivative of cos is just -sin. Why would there be a d[tex]\theta[/tex]/dt at the end?
 
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  • #2
TheDoorsOfMe said:
. But, the derivative of cos is just -sin. Why would there be a d[tex]\theta[/tex]/dt at the end?

No, [itex]\frac{d}{d\theta}\cos\theta=-\sin\theta[/itex] but [itex]\frac{d}{dt}\cos\theta=\left(\frac{d}{d\theta}\cos\theta\right)\left(\frac{d\theta}{dt}\right)[/itex] via the chain rule. :wink:
 
  • #3
TheDoorsOfMe said:

Homework Statement



r = r(t)
[tex]\theta[/tex] = [tex]\theta[/tex](t)

x = r cos([tex]\theta[/tex])

dx/dt =dr/dt cos([tex]\theta[/tex]) - r sin([tex]\theta[/tex]) d[tex]\theta[/tex]/dt





The Attempt at a Solution



Where does the d[tex]\theta[/tex]/dt come from at the end of the derivative? I know I'm using product rule here because r and theta are both functions of t. But, the derivative of cos is just -sin. Why would there be a d[tex]\theta[/tex]/dt at the end?
Chain rule.
d/dt(cos(theta)) = -sin(theta)*d(theta)/dt
 
  • #4
oooooooooooooohhhhhhhhhhhh! man I'm kinda disappointed I didn't see that one : ( oh well. Thank very much guys!
 

What is the Product Rule of x = r cos()?

The Product Rule of x = r cos() is a mathematical rule used to find the derivative of a product of two functions. It states that the derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.

How is the Product Rule applied to x = r cos()?

To apply the Product Rule to x = r cos(), we first identify the two functions being multiplied together: f(x) = r and g(x) = cos(x). Then, we take the derivative of each function separately. The derivative of r is 1 and the derivative of cos(x) is -sin(x). Finally, we plug these values into the Product Rule formula to find the derivative of x = r cos().

Why is the Product Rule important in calculus?

The Product Rule is important in calculus because it allows us to find the derivative of a product of functions, which is a common occurrence in many real-world applications. It is also a fundamental rule that is used to derive other important rules in calculus, such as the Quotient Rule and the Chain Rule.

Can the Product Rule be applied to more than two functions?

Yes, the Product Rule can be applied to any number of functions being multiplied together. For example, if we have three functions f(x), g(x), and h(x), the Product Rule would be: (f(x)g(x)h(x))' = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x).

Are there any exceptions to the Product Rule?

Yes, there are some exceptions to the Product Rule. For example, if one of the functions is a constant, it can be pulled out of the derivative. Additionally, if the two functions being multiplied together are the same, the Product Rule simplifies to the Power Rule.

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