Understanding the Product Rule in Calculus: Solving for y = x ln x

In summary, the conversation discusses finding the derivative of y = x ln x using the product rule. One participant suggests using the known derivative of lnx, 1/x, but another clarifies that product rule should be used to find the derivative of x lnx. The conversation ends with the participants understanding how to use product rule to find the derivative.
  • #1
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y = x ln x

[tex]\frac{dy}{dx} = \frac{1}{x ln x} \cdot 1[/tex]

Is that correct?
 
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  • #2
wrong... use product rule

[tex] \frac{d}{dx} (fg) = \frac{df}{dx}g+f\frac{dg}{dx} [/tex]
 
  • #3
And how would that look like? Which is f & which one is g? :confused:
 
  • #4
Hmm I'm not great at this but,

y = x lnx
I was gunna say.. that whenever you take the derivative of logs.. 1/function * derivative of that function
so that'd give you
y = (1/xlnx)(1/x)
but by using the product rule...
y = 1(xlnx) + x(1/x))
Hmm, definitely don't listen to me, i don't know WHats going on. Here to learn!
 
  • #5
SphericalStrife said:
Hmm I'm not great at this but,

y = x lnx
I was gunna say.. that whenever you take the derivative of logs.. 1/function * derivative of that function
so that'd give you
y = (1/xlnx)(1/x)
I was taught that way. How did you get 1/x?
 
  • #6
hmm I think i might just be being stupid. 1/x is the known derivative of lnx.. but i guess with that other x there.. x lnx, you would need to use product rule like that guy said?
 
  • #7
Oh I see...
SphericalStrife said:
1/x is the known derivative of lnx
How do you know it?
 
  • #8
Use product rule. f(x)=x and g(x)=ln(x)
Now just find derivatives
y = x'*lnx + x*ln(x)'
 
  • #9
Yeah I got it. Thank you.
 

What is the product rule in calculus?

The product rule is a formula used in calculus to find the derivative of two functions that are multiplied together. 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.

When should the product rule be applied?

The product rule should be applied when you have a function that can be broken down into two or more functions multiplied together. It is used to find the derivative of the overall function.

How do you use the product rule in calculus?

To use the product rule, you first need to identify the two functions that are being multiplied together. Then, you apply the formula by taking the derivative of each function and multiplying it by the other function, and then adding those two terms together.

Can the product rule be used for more than two functions?

Yes, the product rule can be extended to more than two functions. For example, if you have three functions multiplied together, you would take the derivative of each function and multiply it by the other two functions, and then add all three terms together.

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 two or more functions. This is a crucial concept in calculus, as it is used in many applications such as optimization, related rates, and curve sketching.

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