Find derivative of a complicated fraction
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Let $n> 1$ be an integer and let $x_1,ldots,x_n$ be positive real numbers, all between $0$ and $1$. Is it possible to find the derivative of it so I know if it is increasing wrt $x_i$?
$$
frac{sum_{i=1}^{n}x_i}{1-prod_{i=1}^{n}(1-x_i)}
$$
I thought about expanding it to $$
frac{x_1+x_2+...x_i}{1-(1-x_1)(1-x_2)...(1-x_i)}
$$
calculus integration multivariable-calculus derivatives partial-derivative
asked Nov 14 at 4:57
Chris
245
add a comment |
up vote
0
down vote
favorite
Let $n> 1$ be an integer and let $x_1,ldots,x_n$ be positive real numbers, all between $0$ and $1$. Is it possible to find the derivative of it so I know if it is increasing wrt $x_i$?
$$
frac{sum_{i=1}^{n}x_i}{1-prod_{i=1}^{n}(1-x_i)}
$$
I thought about expanding it to $$
frac{x_1+x_2+...x_i}{1-(1-x_1)(1-x_2)...(1-x_i)}
$$
calculus integration multivariable-calculus derivatives partial-derivative
asked Nov 14 at 4:57
Chris
245
What have you tried? Have you studied partial derivatives yet?
– DreamConspiracy
Nov 14 at 5:00
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $n> 1$ be an integer and let $x_1,ldots,x_n$ be positive real numbers, all between $0$ and $1$. Is it possible to find the derivative of it so I know if it is increasing wrt $x_i$?
$$
frac{sum_{i=1}^{n}x_i}{1-prod_{i=1}^{n}(1-x_i)}
$$
I thought about expanding it to $$
frac{x_1+x_2+...x_i}{1-(1-x_1)(1-x_2)...(1-x_i)}
$$
calculus integration multivariable-calculus derivatives partial-derivative
asked Nov 14 at 4:57
Chris
245
Let $n> 1$ be an integer and let $x_1,ldots,x_n$ be positive real numbers, all between $0$ and $1$. Is it possible to find the derivative of it so I know if it is increasing wrt $x_i$?
$$
frac{sum_{i=1}^{n}x_i}{1-prod_{i=1}^{n}(1-x_i)}
$$
I thought about expanding it to $$
frac{x_1+x_2+...x_i}{1-(1-x_1)(1-x_2)...(1-x_i)}
$$
calculus integration multivariable-calculus derivatives partial-derivative
calculus integration multivariable-calculus derivatives partial-derivative
asked Nov 14 at 4:57
Chris
245
asked Nov 14 at 4:57
Chris
245
edited Nov 14 at 5:05
asked Nov 14 at 4:57
Chris
245
asked Nov 14 at 4:57
Chris
245
asked Nov 14 at 4:57
Chris
245
245
What have you tried? Have you studied partial derivatives yet?
– DreamConspiracy
Nov 14 at 5:00
add a comment |
What have you tried? Have you studied partial derivatives yet?
– DreamConspiracy
Nov 14 at 5:00
What have you tried? Have you studied partial derivatives yet?
– DreamConspiracy
Nov 14 at 5:00
What have you tried? Have you studied partial derivatives yet?
– DreamConspiracy
Nov 14 at 5:00
add a comment |
1 Answer
1
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oldest
votes
up vote
1
down vote
Hint: Let
$$A=sum_{jneq i}x_j$$
$$B=prod_{jneq i}(1-x_j)$$
answered Nov 14 at 5:02
DreamConspiracy
8261216
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint: Let
$$A=sum_{jneq i}x_j$$
$$B=prod_{jneq i}(1-x_j)$$
answered Nov 14 at 5:02
DreamConspiracy
8261216
add a comment |
up vote
1
down vote
Hint: Let
$$A=sum_{jneq i}x_j$$
$$B=prod_{jneq i}(1-x_j)$$
answered Nov 14 at 5:02
DreamConspiracy
8261216
add a comment |
up vote
1
down vote
up vote
1
down vote
Hint: Let
$$A=sum_{jneq i}x_j$$
$$B=prod_{jneq i}(1-x_j)$$
answered Nov 14 at 5:02
DreamConspiracy
8261216
Hint: Let
$$A=sum_{jneq i}x_j$$
$$B=prod_{jneq i}(1-x_j)$$
answered Nov 14 at 5:02
DreamConspiracy
8261216
answered Nov 14 at 5:02
DreamConspiracy
8261216
answered Nov 14 at 5:02
DreamConspiracy
8261216
answered Nov 14 at 5:02
DreamConspiracy
8261216
8261216
add a comment |
add a comment |
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What have you tried? Have you studied partial derivatives yet?
– DreamConspiracy
Nov 14 at 5:00