Math Problem Statement
nth derivative of 1/(1-5x+6x2)
Solution
The problem asks us to find the th derivative of the function:
Let's break down the steps:
1. Factor the Denominator
First, note that the denominator is a quadratic expression. Let's factor it:
So, the function can be rewritten as:
2. Express as a Partial Fraction
We can express the function as a sum of partial fractions:
where and are constants to be determined. To find and , we equate:
Multiplying through by the denominator gives:
Expanding this expression:
Grouping terms gives:
For this to hold for all , the coefficients of and the constant term must be equal:
- .
- .
Solving these equations simultaneously:
From , we get .
Substituting into :
Then, .
So, the partial fraction decomposition is:
3. Express as a Series
Each term can be expressed as a geometric series:
Applying this to our expression:
Simplifying:
4. Differentiate to Find the th Derivative
The th derivative of is given by:
\frac{k!}{(k-n)!} x^{k-n} & \text{if } k \geq n, \\ 0 & \text{if } k < n. \end{cases}$$ So, the $$n$$th derivative of $$f(x)$$ is: $$f^{(n)}(x) = \sum_{k=n}^{\infty} \left[ -2 \cdot 2^k + 3 \cdot 3^k \right] \frac{k!}{(k-n)!} x^{k-n}.$$ Evaluating this at $$x = 0$$ gives: $$f^{(n)}(0) = \sum_{k=n}^{\infty} \left[ -2 \cdot 2^k + 3 \cdot 3^k \right] \frac{k!}{(k-n)!}.$$ This expression gives the $$n$$th derivative of the function at $$x = 0$$. Would you like more details or have any other questions? Here are some related questions: 1. How do you compute the second derivative of a function? 2. What is the radius of convergence for a geometric series? 3. How do you find the partial fraction decomposition of a function? 4. How do you compute the series expansion for a function? 5. What are the applications of geometric series in calculus? **Tip:** In partial fractions, always check for factors in the denominator that can simplify your problem before expanding.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Differentiation
Partial fractions
Series expansion
Geometric series
Formulas
Geometric series sum formula
Derivative of a power function
Theorems
Binomial theorem
Suitable Grade Level
Advanced Undergraduate