Implicit Differentiation- Related Rates
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Suppose that price, $p$ in dollars, and the number of items sold, $x,$ are related by: $p^2 - xp + x^2 = 175.$
If $p$ and $x$ are functions of time, how fast is $p$ changing with respect to time when $p=$ 10$ and $x$ is increasing by $5$ units per day?
So far I’ve got: $$fracdpdt = frac-2x+p2p-x.$$
Not sure what to do after this? Help would be extremely appreciated :(
calculus multivariable-calculus implicit-differentiation
edited Aug 8 at 16:38
Adrian Keister
3,61321533
asked Aug 8 at 16:28
NotAMathWizard
84
add a comment |Â
up vote
1
down vote
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Suppose that price, $p$ in dollars, and the number of items sold, $x,$ are related by: $p^2 - xp + x^2 = 175.$
If $p$ and $x$ are functions of time, how fast is $p$ changing with respect to time when $p=$ 10$ and $x$ is increasing by $5$ units per day?
So far I’ve got: $$fracdpdt = frac-2x+p2p-x.$$
Not sure what to do after this? Help would be extremely appreciated :(
calculus multivariable-calculus implicit-differentiation
edited Aug 8 at 16:38
Adrian Keister
3,61321533
asked Aug 8 at 16:28
NotAMathWizard
84
NotAMathWizard, your solution for $dp over dt$ is not quite correct. It should be $ frac dpdt = frac dxdt frac -2x+p2p-x$. (See Adrian Keister's answer below.) Also, in case the notation is unfamiliar to you, Adrian Keister used a "dot notation" to indicate the derivatives of $x$ and $p$. My understanding is that is popular with physicists to indicate a derivative with respect to time.
– Randall Blake
Aug 10 at 17:02
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose that price, $p$ in dollars, and the number of items sold, $x,$ are related by: $p^2 - xp + x^2 = 175.$
If $p$ and $x$ are functions of time, how fast is $p$ changing with respect to time when $p=$ 10$ and $x$ is increasing by $5$ units per day?
So far I’ve got: $$fracdpdt = frac-2x+p2p-x.$$
Not sure what to do after this? Help would be extremely appreciated :(
calculus multivariable-calculus implicit-differentiation
edited Aug 8 at 16:38
Adrian Keister
3,61321533
asked Aug 8 at 16:28
NotAMathWizard
84
Suppose that price, $p$ in dollars, and the number of items sold, $x,$ are related by: $p^2 - xp + x^2 = 175.$
If $p$ and $x$ are functions of time, how fast is $p$ changing with respect to time when $p=$ 10$ and $x$ is increasing by $5$ units per day?
So far I’ve got: $$fracdpdt = frac-2x+p2p-x.$$
Not sure what to do after this? Help would be extremely appreciated :(
calculus multivariable-calculus implicit-differentiation
edited Aug 8 at 16:38
Adrian Keister
3,61321533
asked Aug 8 at 16:28
NotAMathWizard
84
edited Aug 8 at 16:38
Adrian Keister
3,61321533
edited Aug 8 at 16:38
Adrian Keister
3,61321533
edited Aug 8 at 16:38
Adrian Keister
3,61321533
3,61321533
asked Aug 8 at 16:28
NotAMathWizard
84
asked Aug 8 at 16:28
NotAMathWizard
84
asked Aug 8 at 16:28
NotAMathWizard
84
84
NotAMathWizard, your solution for $dp over dt$ is not quite correct. It should be $ frac dpdt = frac dxdt frac -2x+p2p-x$. (See Adrian Keister's answer below.) Also, in case the notation is unfamiliar to you, Adrian Keister used a "dot notation" to indicate the derivatives of $x$ and $p$. My understanding is that is popular with physicists to indicate a derivative with respect to time.
– Randall Blake
Aug 10 at 17:02
add a comment |Â
NotAMathWizard, your solution for $dp over dt$ is not quite correct. It should be $ frac dpdt = frac dxdt frac -2x+p2p-x$. (See Adrian Keister's answer below.) Also, in case the notation is unfamiliar to you, Adrian Keister used a "dot notation" to indicate the derivatives of $x$ and $p$. My understanding is that is popular with physicists to indicate a derivative with respect to time.
– Randall Blake
Aug 10 at 17:02
NotAMathWizard, your solution for $dp over dt$ is not quite correct. It should be $ frac dpdt = frac dxdt frac -2x+p2p-x$. (See Adrian Keister's answer below.) Also, in case the notation is unfamiliar to you, Adrian Keister used a "dot notation" to indicate the derivatives of $x$ and $p$. My understanding is that is popular with physicists to indicate a derivative with respect to time.
– Randall Blake
Aug 10 at 17:02
NotAMathWizard, your solution for $dp over dt$ is not quite correct. It should be $ frac dpdt = frac dxdt frac -2x+p2p-x$. (See Adrian Keister's answer below.) Also, in case the notation is unfamiliar to you, Adrian Keister used a "dot notation" to indicate the derivatives of $x$ and $p$. My understanding is that is popular with physicists to indicate a derivative with respect to time.
– Randall Blake
Aug 10 at 17:02
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Let's see:
beginalign*
fracddt[,p^2-xp+x^2&=175,] \
underbrace2pdotp_textChain-underbrace(xdotp+dotxp)_textProduct+underbrace2xdotx_textChain&=0 \
dotp(2p-x)&=dotx(p-2x) \
dotp&=fracdotx(p-2x)2p-x.
endalign*
The issue is that the problem statement doesn't give you the value of $x$, but of $dotx.$ You can find $x$ by solving the original equation for it when you've plugged in $p=10.$ That is, you are solving
$$100-10x+x^2=175,qquadtextorqquad x^2-10x-75=0.$$
The solutions are $x=-5, 15.$ Can you rule out one of these? Why?
answered Aug 8 at 16:44
Adrian Keister
3,61321533
The question doesn’t give more detail other then the fact it’s a implicit differentiation question... I’m just stuck where to go.
– NotAMathWizard
Aug 8 at 19:24
Ok, see the edit. Can you continue?
– Adrian Keister
Aug 8 at 20:39
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Let's see:
beginalign*
fracddt[,p^2-xp+x^2&=175,] \
underbrace2pdotp_textChain-underbrace(xdotp+dotxp)_textProduct+underbrace2xdotx_textChain&=0 \
dotp(2p-x)&=dotx(p-2x) \
dotp&=fracdotx(p-2x)2p-x.
endalign*
The issue is that the problem statement doesn't give you the value of $x$, but of $dotx.$ You can find $x$ by solving the original equation for it when you've plugged in $p=10.$ That is, you are solving
$$100-10x+x^2=175,qquadtextorqquad x^2-10x-75=0.$$
The solutions are $x=-5, 15.$ Can you rule out one of these? Why?
answered Aug 8 at 16:44
Adrian Keister
3,61321533
The question doesn’t give more detail other then the fact it’s a implicit differentiation question... I’m just stuck where to go.
– NotAMathWizard
Aug 8 at 19:24
Ok, see the edit. Can you continue?
– Adrian Keister
Aug 8 at 20:39
add a comment |Â
up vote
2
down vote
accepted
Let's see:
beginalign*
fracddt[,p^2-xp+x^2&=175,] \
underbrace2pdotp_textChain-underbrace(xdotp+dotxp)_textProduct+underbrace2xdotx_textChain&=0 \
dotp(2p-x)&=dotx(p-2x) \
dotp&=fracdotx(p-2x)2p-x.
endalign*
The issue is that the problem statement doesn't give you the value of $x$, but of $dotx.$ You can find $x$ by solving the original equation for it when you've plugged in $p=10.$ That is, you are solving
$$100-10x+x^2=175,qquadtextorqquad x^2-10x-75=0.$$
The solutions are $x=-5, 15.$ Can you rule out one of these? Why?
answered Aug 8 at 16:44
Adrian Keister
3,61321533
The question doesn’t give more detail other then the fact it’s a implicit differentiation question... I’m just stuck where to go.
– NotAMathWizard
Aug 8 at 19:24
Ok, see the edit. Can you continue?
– Adrian Keister
Aug 8 at 20:39
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Let's see:
beginalign*
fracddt[,p^2-xp+x^2&=175,] \
underbrace2pdotp_textChain-underbrace(xdotp+dotxp)_textProduct+underbrace2xdotx_textChain&=0 \
dotp(2p-x)&=dotx(p-2x) \
dotp&=fracdotx(p-2x)2p-x.
endalign*
The issue is that the problem statement doesn't give you the value of $x$, but of $dotx.$ You can find $x$ by solving the original equation for it when you've plugged in $p=10.$ That is, you are solving
$$100-10x+x^2=175,qquadtextorqquad x^2-10x-75=0.$$
The solutions are $x=-5, 15.$ Can you rule out one of these? Why?
answered Aug 8 at 16:44
Adrian Keister
3,61321533
Let's see:
beginalign*
fracddt[,p^2-xp+x^2&=175,] \
underbrace2pdotp_textChain-underbrace(xdotp+dotxp)_textProduct+underbrace2xdotx_textChain&=0 \
dotp(2p-x)&=dotx(p-2x) \
dotp&=fracdotx(p-2x)2p-x.
endalign*
The issue is that the problem statement doesn't give you the value of $x$, but of $dotx.$ You can find $x$ by solving the original equation for it when you've plugged in $p=10.$ That is, you are solving
$$100-10x+x^2=175,qquadtextorqquad x^2-10x-75=0.$$
The solutions are $x=-5, 15.$ Can you rule out one of these? Why?
answered Aug 8 at 16:44
Adrian Keister
3,61321533
edited Aug 8 at 20:39
answered Aug 8 at 16:44
Adrian Keister
3,61321533
answered Aug 8 at 16:44
Adrian Keister
3,61321533
answered Aug 8 at 16:44
Adrian Keister
3,61321533
3,61321533
The question doesn’t give more detail other then the fact it’s a implicit differentiation question... I’m just stuck where to go.
– NotAMathWizard
Aug 8 at 19:24
Ok, see the edit. Can you continue?
– Adrian Keister
Aug 8 at 20:39
add a comment |Â
The question doesn’t give more detail other then the fact it’s a implicit differentiation question... I’m just stuck where to go.
– NotAMathWizard
Aug 8 at 19:24
Ok, see the edit. Can you continue?
– Adrian Keister
Aug 8 at 20:39
The question doesn’t give more detail other then the fact it’s a implicit differentiation question... I’m just stuck where to go.
– NotAMathWizard
Aug 8 at 19:24
The question doesn’t give more detail other then the fact it’s a implicit differentiation question... I’m just stuck where to go.
– NotAMathWizard
Aug 8 at 19:24
Ok, see the edit. Can you continue?
– Adrian Keister
Aug 8 at 20:39
Ok, see the edit. Can you continue?
– Adrian Keister
Aug 8 at 20:39
add a comment |Â
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NotAMathWizard, your solution for $dp over dt$ is not quite correct. It should be $ frac dpdt = frac dxdt frac -2x+p2p-x$. (See Adrian Keister's answer below.) Also, in case the notation is unfamiliar to you, Adrian Keister used a "dot notation" to indicate the derivatives of $x$ and $p$. My understanding is that is popular with physicists to indicate a derivative with respect to time.
– Randall Blake
Aug 10 at 17:02