Finding the nth derivative of f(x) = x/(x+1)

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Start date Apr 24, 2020
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Derivative

In summary, the formula for finding the nth derivative of f(x) = x/(x+1) is n!/(x+1)^(n+1). To find the first derivative, use the power rule and quotient rule: f'(x) = [(x+1)(1) - x(1)]/(x+1)^2. The second derivative is the derivative of the first derivative, f''(x) = -2/(x+1)^3. The nth derivative will have n+1 terms, with the coefficients following the pattern: 1, -2, 6, -24, ... where the coefficient is given by (-1)^(n+1) * n!.

Apr 24, 2020

ttpp1124

Homework Statement: Finding the nth derivative.

Relevant Equations: n/a

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Last edited by a moderator: Apr 24, 2020

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Apr 24, 2020

benorin

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1. What is the formula for finding the nth derivative of f(x) = x/(x+1)?

The formula for finding the nth derivative of f(x) = x/(x+1) is (-1)^n * n! / (x+1)^(n+1).

2. How do you find the first derivative of f(x) = x/(x+1)?

To find the first derivative, we use the power rule and quotient rule. First, we rewrite the function as f(x) = x * (x+1)^(-1). Then, using the power rule, we get f'(x) = 1 * (x+1)^(-1) + x * (-1) * (x+1)^(-2). Simplifying, we get f'(x) = (x+1)^(-2) - x(x+1)^(-2). Finally, using the quotient rule, we get f'(x) = (x+1)^(-2) * (1 - x).

3. How many terms are in the nth derivative of f(x) = x/(x+1)?

The nth derivative of f(x) = x/(x+1) will have n+1 terms.

4. Can the nth derivative of f(x) = x/(x+1) be simplified?

Yes, the nth derivative can be simplified using algebraic manipulations and the properties of derivatives. For example, the second derivative can be simplified to f''(x) = 2(x+1)^(-3) * (1 - 2x).

5. How does the value of x affect the nth derivative of f(x) = x/(x+1)?

The value of x affects the nth derivative of f(x) = x/(x+1) in that it determines the coefficients of each term. As x increases or decreases, the coefficients will change accordingly, resulting in a different nth derivative.