All Questions
Tagged with partial-fractions derivatives
21
questions
0
votes
1
answer
69
views
If the third derivative of $\frac{x^4}{(x-1)(x-2)}$ is $\frac{-12k}{(x-2)^4}$ + $\frac{6}{(x-1)^4}$ then the value of k is? [closed]
In the answers I found on google, see this link, they converted the given function into a certain form? What is the process of that conversion (I understand it is a partial fraction of sorts, but how ...
0
votes
1
answer
62
views
Higher order partial derivatives of a product of reciprocals of cumulative sums
This question is related to another question where we asked for evaluating a certain multi-dimensional integral. It turns out that that integral, from the question above, can be reduced to an action ...
2
votes
2
answers
120
views
How to decompose $\frac{1+x}{\sqrt{(1-x)}}$ into partial fractions?
Basically homework help. The question (Problems of Calculus in One Variable, IA Maron, number 2.3.9(b)) is to find the derivative of the 100th order of the function
$$
y = \frac{1+x}{\sqrt{(1-x)}}
$$
...
0
votes
2
answers
179
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Finding nth derivative of $\frac{1}{x^4+4}$
I am supposed to find the nth order derivative of:
$$\frac{1}{x^4+4}$$
I tried to resolve into partial fractions. But it didn't work out for me.
Edit- where I am stuck
$$\frac{1}{x^4+4}=\frac{1}{(x-1+...
0
votes
1
answer
53
views
Decompose into simple fractions $\frac{f'}{f}$
Let $f(x) = (x-a_1)(x-a_2)...(x-a_n)$.
Find a decomposition into simple fractions of $\frac{f'}{f}$.
Where $f'$ is a derivative of our polynomial.
As I understand, we have to find a pretty-format of $...
9
votes
3
answers
278
views
Compute 100th derivative [duplicate]
A friend suggested me a rather tricky problem, namely find the $100^{th}$ derivative of
$$
f(x)=\frac{x^2+1}{x^3-x}.
$$
I have computed the zeroth derivative
$$
\frac{x^2+1}{x^3-x}
$$
and the first ...
3
votes
2
answers
88
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$f(x) = \frac{4 + x}{2 + x - x^2}$, calculate $f^{(9)}(1)$
$f(x) = \frac{4 + x}{2 + x - x^2}$, calculate $f^{(9)}(1)$, where $f^{(9)}$ is the $9$-th derivative of $f$.
Domain of $f$ is $\mathbb{R} - \{-1, 2\}$. I've got that $f(x) = \frac{1}{1 - (-x)} + \...
0
votes
1
answer
39
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$2n$ th partial derivative of $\frac{1}{y(1+x^2)-1}$ with respect to $x$.
I need to find the $2n$ th derivative with respect to $x$ of the function $f = \frac{1}{y(1+x^2)-1}$. I tried differentiating util a pattern was founded, but that didn't happen.
I think the $x^2$ is ...
10
votes
1
answer
156
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A determinant involving a polynomial is $0$
Let $n \geq 2$ and $f:\mathbb{R} \to \mathbb{R}, \: f(x)=(x-x_1)(x-x_2)\dots(x-x_n)$ where $x_1,\dots, x_n$ are distinct real numbers. The matrix $A=(a_{ij})_{1 \leq i,j \leq n}$ is defined as follows:...
2
votes
3
answers
116
views
Formula for $1/f(x)$ where $f$ is a polynomial
Let $f$ be a polynomial having $n$ distinct real roots:
$$f(x)=(x-x_1)(x-x_2)\dots(x-x_n)$$
Prove that $$\frac{1}{f(x)}=\sum_{k=1}^n \frac{1}{f'(x_k)(x-x_k)}, \: \forall x \in \mathbb{R} - \{x_1,...
0
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0
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139
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How do we compute higher order derivatives of a rational function?
Let $a \in (-1,1)$ and $x\in {\mathbb R}$ and consider a following sequence of rational functions:
\begin{eqnarray}
R^{(0)}[x]&:=& \frac{x (a x-1)}{a x-x^2+x-1}\\
R^{(1)}[x]&:=& -\frac{...
0
votes
0
answers
34
views
Fractional derivatives in different coordinates
I wonder how fractional derivatives are defined in cylindrical coordinate?
Does anyone have any idea?
4
votes
0
answers
570
views
Find the derivative of integral $f(x)/(x^2(x-5)^7)$ when $f(x)$ is a quadratic function.
This question is quite tricky. It's for my Calculus 2 assignment and I can't seem to figure out how to integrate this function in order to get its derivative. I tried partial fractions, u-sub with x-5 ...
0
votes
2
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623
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Natural parameterization of the following curves:
I am having trouble finding the natural parameterization of these curves:
$$\alpha(t)=\left(\sin^2\left(\frac{t}{\sqrt{2}}\right),\frac{1}{2}\sin \left(t\sqrt{2}\right), \left(\frac{t}{\sqrt{2}}\...
0
votes
1
answer
37
views
How can I find derivatives of this function with respect to $U_i$ and $V_j$?
I have this objective function, I want to find its derivative with respect to $U_i$ and $V_j$, I don't know how should I approach these kind of functions, it would be helpful if any one could tell me ...
0
votes
1
answer
204
views
Partial Fraction Expansion and Laplace Transform
I have some problem with my equation:
$$y''+5y = 5e^{-5t} $$
And we got that y(0) = 1 and y'(0) = 2
It's too much to write but what I get is:
$$ \frac{e^{-5t}}{6}+1.516676089\cdot \cos(0.989 - \sqrt{...
0
votes
2
answers
55
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Unsustainable Populations Differential Equations
I need some help solving this differential equation: $\frac{\text{d}P}{\text{d}t}=kP\left(\frac{A-P}{A}\right)\left(\frac{P-m}{P}\right)$, where $P$ is the population, $t$ is time in years, $A$ is the ...
2
votes
3
answers
73
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Calculating the $n^\text{th}$ derivative
How do we calculate the $n^{\text{th}}$ derivative for
$$
\frac{x^3}{(x-a)(x-b)(x-c)}?
$$
How can I obtain the partial fraction for the given term?
1
vote
0
answers
57
views
Determine lambda from a non-constant differentiable function of one variable
Suppose f is a non-constant differentiable function of one variable.
Determine, with reasons, the value of $\lambda$ for which
F(x, y) = f($\lambda x^{3}$ + y) satisfies the partial differential ...
1
vote
4
answers
11k
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How to find nth derivative of $1/(1+x+x^2+x^3)$
I was trying to solve a differentiation question but unable to understand .
My question is :
find the $n^{th}$ derivative of $1/(1+x+x^2+x^3)$
I know that if we divide the numerator by denominator ...
1
vote
2
answers
865
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Partial fraction when $N^r$ and $D^r$ are quadratic and cubic polynomials
I need to find the nth derivative of the following function
$$y=\frac {x^2+4x+1}{x^3+2x^2-x-2}$$
The trouble is I don't know how to break a fraction like the above one. How do I break it into partial ...