What is wrong with this differentiation?

In summary, the conversation is about finding the derivative dy/dt for a given function y=x^2-5x+1 and variables x=s^3-2s+1 and s=\sqrt{t^2+1}. The conversation includes an attempt at using the chain rule, but the resulting answer is not correct. After doing some manipulations, it is discovered that both answers are actually equal.
  • #1
pc2-brazil
205
3

Homework Statement


Find dy/dt, given:
[tex]y=x^2-5x+1[/tex]
[tex]x=s^3-2s+1[/tex]
[tex]s=\sqrt{t^2+1}[/tex]

2. The attempt at a solution
I am trying to use the chain rule for the case where y is a function of x, x is a function of s and s is a function of t, like below:
[tex]\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{ds}\frac{ds}{dt}[/tex]
So:
[tex]\frac{dy}{dt}=(2x-5)(3s^2-2)(\frac{t}{\sqrt{t^2+1}})[/tex]
But this gives (substituting x and s for their values as functions of t)
[tex]6t^5-4t^3-\frac{3t}{\sqrt{t^2+1}}-\frac{9t^3}{\sqrt{t^2+1}}-2t[/tex]
which is not the correct result. The correct result is:
[tex]6t^5-4t^3-9t\sqrt{t^2+1}+\frac{6t}{\sqrt{t^2+1}}-2t[/tex]
What is wrong?

Thank you in advance.
 
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  • #2
Never mind. After doing some manipulations I found that apparently both results are equal, because these two terms of the answer given as correct:
[tex]-9t\sqrt{t^2+1}+\frac{6t}{\sqrt{t^2+1}}[/tex]
after some manipulation:
[tex]-9t\sqrt{t^2+1}+\frac{6t}{\sqrt{t^2+1}}=\frac{-9t(t^2+1)+6t}{\sqrt{t^2+1}}=\frac{-9t^3-3t}{\sqrt{t^2+1}}[/tex]
become the two terms of the answer I found.
 
Last edited:

Related to What is wrong with this differentiation?

1. What is differentiation?

Differentiation is a mathematical process used to find the rate of change of a function with respect to its independent variable. It is the inverse of integration and is commonly used to solve problems in calculus, physics, and economics.

2. Why is differentiation important?

Differentiation is important because it allows us to analyze the behavior of a function and understand how it changes over time or in response to certain variables. This is crucial in many scientific fields, as it helps us make predictions, optimize processes, and find solutions to complex problems.

3. What can go wrong with differentiation?

There are several things that can go wrong with differentiation, such as applying the wrong formula, making mistakes in the algebraic manipulation of the function, or not considering the differentiability of the function at certain points. These errors can lead to incorrect results and must be carefully avoided.

4. How can I tell if my differentiation is correct?

To ensure the correctness of your differentiation, you can check your work using various methods such as graphing the function and its derivative, using the chain rule to verify the result, or comparing it to a known solution. It is also helpful to double-check your algebra and calculations for any mistakes.

5. Can differentiation be used in all situations?

No, differentiation cannot be used in all situations. It is limited to functions that are continuous and differentiable, meaning that they have no breaks or sharp turns and can be smoothly traced with a single line. In some cases, such as for piecewise functions or functions with discontinuities, other methods may be needed.

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