Solving Calculus Question f(x) | Viet Dao

• VietDao29
Viet Dao was trying to find the rule for f^{(n)}(x), but was having difficulty. They were able to find f'(x) and f''(x), but f^{(n)}(x) was too complicated. They asked for help and Daniel provided a formula for f^{(n)}(x) using induction. Viet Dao then asked if their formula was correct and Daniel confirmed that it was. In summary, Viet Dao was struggling to find the rule for f^{(n)}(x) and asked for help. Daniel provided a formula using induction and confirmed that it was correct.
VietDao29
Homework Helper
Hi,
$$f(x) = \frac{5x^{2} - 3x - 20}{x^{2} - 2x - 3}$$
Find:
$$f^{(n)}(x)$$
I try to find its rule, but I fail:
I have:
$$f'(x) = \frac{-7x^{2} + 10x - 31}{(x^{2} - 2x - 3)^{2}}$$
And I have f''(x) is some kind of very very complicated number. What should I do in this kind of problem??
Any help will be appreciated,
Viet Dao,

Last edited:
Differentiate a few times simplifying as much as you can. Here is what I got:

$$f(x) = 5 + \frac{4}{x - 3} + \frac{3}{x + 1}$$

$$f'(x) = -\frac{4}{(x - 3)^2} - \frac{3}{(1 + x)^2}$$

I feel if I go any further I'll completely give the pattern away, carry on differentiating.

$$f^{(n)} = \frac{(-1)^{n}4(n!)}{(x - 3)^{n + 1}} + \frac{(-1)^{n}3(n!)}{(x + 1)^{n + 1}}$$
Am I correct??

U can use induction method to prove your formula right...

Daniel.

So...
$$f^{1}(x) = -\frac{4}{(x - 3)^{2}} - \frac{3}{(1 + x)^2}$$
So the formula is correct if n = 1.
Assume it's correct for n = k:
$$f^{(k)}(x) = \frac{(-1)^{k}4(k!)}{(x - 3)^{k + 1}} + \frac{(-1)^{k}3(k!)}{(x + 1)^{k + 1}}$$
Prove it's correct if n = k + 1:
$$f^{(k + 1)}(x) = 4(k!)(-1)^{k}\frac{-(k + 1)(x - 3)^{k}}{(x - 3)^{2k + 2}} + (-1)^{k}3(k!)\frac{-(k + 1)(x + 1)^{k}}{(x + 1)^{2k + 2}}$$
$$= \frac{4(k + 1)!(-1)^{k + 1}}{(x - 3)^{k + 2}} + \frac{(-1)^{k + 1}3(k + 1)!}{(x + 1)^{k + 2}}$$
So the formula is true $\forall n \in N*$
Am I correct?
Viet Dao,

Last edited:

1. What is Calculus?

Calculus is a branch of mathematics that deals with the study of change and motion by using mathematical models and techniques, such as derivatives and integrals.

2. What is a function in Calculus?

In Calculus, a function is a mathematical rule that maps a set of inputs to a set of outputs. It represents the relationship between the input and output variables and can be represented by a graph, equation, or table.

3. How do I use the f(x) notation in calculus?

The f(x) notation is used to represent a function, where f is the name of the function and x is the input variable. It is commonly used in calculus to represent the dependent variable (y) in terms of the independent variable (x).

4. What is the process for solving a calculus question with f(x)?

To solve a calculus question with f(x), you will first need to understand the problem and identify the given information. Then, you can use mathematical techniques, such as derivatives or integrals, to find the solution. Finally, you will need to check your answer and make sure it satisfies the given conditions.

5. What are some common applications of using f(x) in calculus?

Some common applications of using f(x) in calculus include finding the rate of change, optimizing functions, and solving real-world problems involving motion, growth, and decay. It is also used in fields such as physics, engineering, and economics to model and analyze various phenomena.

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