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There was some confusion about whether I originally meant left or right so I made it more explicit.
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Thierry
Thierry
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I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, and maybe the students can see when that happens). Let's call this function $g$.

Then, using your example, I would rewrite $x+xy+y^2=7$ as $x+xg(x)+g(x)^2=7$, and emphasize that it holds locally. The right hand sideSince $x+xg(x)+g(x)^2$ is a function of $x$ and it's constant function in our little region, so its derivative is zero there. Now we just use basic differentiation rules to get an expression for $g'$, and thus the tangent line.

I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, and maybe the students can see when that happens). Let's call this function $g$.

Then, using your example, I would rewrite $x+xy+y^2=7$ as $x+xg(x)+g(x)^2=7$, and emphasize that it holds locally. The right hand side is a function of $x$ and it's constant, so its derivative is zero. Now we just use basic differentiation rules to get an expression for $g'$, and thus the tangent line.

I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, and maybe the students can see when that happens). Let's call this function $g$.

Then, using your example, I would rewrite $x+xy+y^2=7$ as $x+xg(x)+g(x)^2=7$, and emphasize that it holds locally. Since $x+xg(x)+g(x)^2$ is a constant function in our little region, its derivative is zero there. Now we just use basic differentiation rules to get an expression for $g'$, and thus the tangent line.

the author made a little mistake by writing left when he / she meant right
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edit approved Sep 30, 2021 at 0:43
Jane N.
edit approved Sep 30, 2021 at 0:43
Jane N.
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I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, and maybe the students can see when that happens). Let's call this function $g$.

Then, using your example, I would rewrite $x+xy+y^2=7$ as $x+xg(x)+g(x)^2=7$, and emphasize that it holds locally. The leftright hand side is a function of $x$ and it's constant, so its derivative is zero. Now we just use basic differentiation rules to get an expression for $g'$, and thus the tangent line.

I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, and maybe the students can see when that happens). Let's call this function $g$.

Then, using your example, I would rewrite $x+xy+y^2=7$ as $x+xg(x)+g(x)^2=7$, and emphasize that it holds locally. The left hand side is a function of $x$ and it's constant, so its derivative is zero. Now we just use basic differentiation rules to get an expression for $g'$, and thus the tangent line.

I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, and maybe the students can see when that happens). Let's call this function $g$.

Then, using your example, I would rewrite $x+xy+y^2=7$ as $x+xg(x)+g(x)^2=7$, and emphasize that it holds locally. The right hand side is a function of $x$ and it's constant, so its derivative is zero. Now we just use basic differentiation rules to get an expression for $g'$, and thus the tangent line.

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Thierry
Thierry
  • 1.6k
  • 7
  • 9

I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, and maybe the students can see when that happens). Let's call this function $g$.

Then, using your example, I would rewrite $x+xy+y^2=7$ as $x+xg(x)+g(x)^2=7$, and emphasize that it holds locally. The left hand side is a function of $x$ and it's constant, so its derivative is zero. Now we just use basic differentiation rules to get an expression for $g'$, and thus the tangent line.