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anil86
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Find nth Derivative - Quick Tutorial
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In summary, the conversation is discussing finding the nth derivative and proving a given equation using induction. The attachment provided was out-of-focus, and the original question did not ask for the nth derivative. Instead, the individual has shown that y_1 = \frac{\gamma y}{1+x} + \frac{\gamma y}{1-x} = \frac{2\gamma y}{1-x^2} and used it as the base case for the induction proof.
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Ackbach
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Could you please re-scan your attachment? The current one is so out-of-focus as to be nearly useless.
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anil86
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Opalg
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The assignment does not ask you to find the $n$th derivative, but to prove that \(\displaystyle (1-x^2)y_{n+1} - 2(\gamma + nx) y_n -n(n-1)y_{n-1} = 0.\)
You have shown that \(\displaystyle y_1 = \frac{\gamma y}{1+x} + \frac{\gamma y}{1-x} = \frac{2\gamma y}{1-x^2}.\) Write that as \(\displaystyle (1-x^2)y_1 - 2\gamma y = 0.\) Differentiate, to get \(\displaystyle (1-x^2)y_2 - 2xy_1 - 2\gamma y_1 = 0.\) Now use that as the base case for a proof by induction.
You have shown that \(\displaystyle y_1 = \frac{\gamma y}{1+x} + \frac{\gamma y}{1-x} = \frac{2\gamma y}{1-x^2}.\) Write that as \(\displaystyle (1-x^2)y_1 - 2\gamma y = 0.\) Differentiate, to get \(\displaystyle (1-x^2)y_2 - 2xy_1 - 2\gamma y_1 = 0.\) Now use that as the base case for a proof by induction.
What is a derivative?
A derivative is a mathematical concept that represents the instantaneous rate of change of a function at a specific point. It is the slope of the tangent line at that point and is used to analyze the behavior of a function.
Why is finding the nth derivative important?
Finding the nth derivative allows us to analyze the behavior of a function at higher levels of precision. It also helps us to understand the rate of change of a function at different points and can be used to solve complex mathematical problems.
How do you find the nth derivative of a function?
To find the nth derivative of a function, you will need to use the power rule, product rule, quotient rule, and chain rule. These rules help you to simplify the function and eventually find the nth derivative by repeatedly differentiating the function.
Are there any shortcuts for finding the nth derivative?
Yes, there are a few shortcuts for finding the nth derivative. One of the most commonly used shortcuts is the Leibniz notation, which uses the notation f(n)(x) to represent the nth derivative of a function. Another shortcut is to use the Taylor series expansion to find the nth derivative.
What are some real-world applications of finding the nth derivative?
Finding the nth derivative has many real-world applications, particularly in the fields of physics, engineering, and economics. It is used to analyze the motion of objects, determine the velocity and acceleration of a moving object, and solve optimization problems.
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