Differentiation of Log(cos(X)/x^2)^2

In summary: Thanks for catching that mistake. In summary, the conversation involves differentiating the expression y = (log(t))^2 using the chain rule, derivative of log, and quotient rule. There is a discussion about the correct way to differentiate and simplify the equation, with one approach using the chain rule and the other using the rules of logs.
  • #1
Anne5632
23
2
Homework Statement
Differentiate
Relevant Equations
Log(cos(X)/x^2)^2
Im going by the chain rule.
I let y=log(t)^2.
T=cos^2x/x^2Dy/DT is 2/t * log(t)
Dt/DX is (sin(2x)/X )((sinx+cosx)/X)
Can someone verify this is the correct way ? As when I multiply dydt by dtdx I get an equation I don't know how to simplify
 
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  • #2
Never mind solved
 
  • #3
Anne5632 said:
Homework Statement:: Differentiate
Relevant Equations:: Log(cos(X)/x^2)^2
This expression (it's not an equation) is somewhat ambiguous. Just to verify, is this what you're working with?
$$y = \left[\log\left( \frac {\cos(x)}{x^2}\right)\right]^2$$
Anne5632 said:
Im going by the chain rule.
I let y=log(t)^2.
T=cos^2x/x^2
You need to use more than just the chain rule. I used, in this order, the chain rule, derivative of log, and quotient rule.
Anne5632 said:
Dy/DT is 2/t * log(t)
(Edited to correct my error)
No. If ##y = (\log(t))^2##, then ##\frac{dy}{dt} = 2\cdot \log(t) \frac d{dt} \log(t) = 2\frac {\log(t)} t##
Also, try to be more consistent in your use of variables. You have x, X, t, T, Dy/DT, dydt, dtdx, Dt/DX.
Anne5632 said:
Dt/DX is (sin(2x)/X )((sinx+cosx)/X)
Can someone verify this is the correct way ? As when I multiply dydt by dtdx I get an equation I don't know how to simplify
I don't get anything close to this.
 
Last edited:
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  • #4
I would start from [tex]
(\log(\cos(x)/x^2))^2 = (\log(\cos(x)) - 2 \log x)^2[/tex] and use [tex]
\frac{d}{dx} g(x)^2 = 2g(x) \frac{dg}{dx}.[/tex]
 
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  • #5
Yea instead of making it hard and doing chain in the beginning I simplified with rules of logs
 
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  • #6
Anne5632 said:
Yea instead of making it hard and doing chain in the beginning I simplified with rules of logs.
You, of course, should be able to do it both ways, and it can be good practice to see how to transform/simplify one answer into the other.

Mark44 said:
No. If ##y = (\log(t))^2##, then ##\frac{dy}{dt} = 2\cdot \frac d{dt} \log(t) = \frac 2 t##
@Anne5632 had it right. You're still going to have a ##(\log t)^1## when you apply the chain rule.

Mark44 said:
I don't get anything close to this.
Neither did I.
 
  • #7
Mark44 said:
No. If ##y = (\log(t))^2##, then ##\frac{dy}{dt} = 2\cdot \frac d{dt} \log(t) = \frac 2 t##
vela said:
@Anne5632 had it right. You're still going to have a ##(\log t)^1## when you apply the chain rule.
You're right. It should have been ##2\cdot \log(t) \cdot \frac d{dt} \log(t) = 2 \frac{\log(t)} t##.
 

FAQ: Differentiation of Log(cos(X)/x^2)^2

What is the formula for Differentiation of Log(cos(X)/x^2)^2?

The formula for Differentiation of Log(cos(X)/x^2)^2 is (2cos(X)sin(X) + 4xln(cos(X)))/(x^3).

How do you differentiate Log(cos(X)/x^2)^2?

To differentiate Log(cos(X)/x^2)^2, you can use the quotient rule and chain rule. First, take the derivative of the numerator and denominator separately. Then, substitute the values back into the quotient rule formula and simplify.

What is the purpose of Differentiation of Log(cos(X)/x^2)^2?

The purpose of Differentiation of Log(cos(X)/x^2)^2 is to find the rate of change of the given function with respect to its independent variable. This can be useful in various applications of mathematics and science, such as optimization problems and modeling real-world phenomena.

Can Differentiation of Log(cos(X)/x^2)^2 be applied to any function?

No, Differentiation of Log(cos(X)/x^2)^2 can only be applied to functions that involve logarithms and trigonometric functions. It cannot be applied to functions that do not contain these elements.

Are there any alternative methods for Differentiation of Log(cos(X)/x^2)^2?

Yes, there are alternative methods for Differentiation of Log(cos(X)/x^2)^2 such as using the product rule or the power rule. However, the quotient rule and chain rule are the most efficient and accurate methods for differentiating this function.

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