Use implicit differentiation to find the slope of the tangent line to the curve defined by

x
y
2
+
2
x
y
=
5
at the point
(
3
,
1
)
.

Question

Use implicit differentiation to find the slope of the tangent line to the curve defined by

x
y
2
+
2
x
y
=
5
at the point 
(
3
,
1
)
.

BlackTom AI · Accepted Answer

We start by clarifying the given equation and the point of tangency.
The curve is defined implicitly by x y^2 + 2 x y = 5, and we are asked for the slope dy/dx at the point (3, 1).
Now we differentiate implicitly with respect to x.
- For the term x y^2, apply the product rule: d/dx[x y^2] = y^2 + x*(2y*y').
- For the term 2 x y, apply the product rule again: d/dx[2 x y] = 2*(y) + 2x*y'.
- The right-hand side derivative of 5 is 0.
Putting these together, we get: y^2 + 2xy y' + 2y + 2x y' = 0.
Next, substitute the coordinates x = 3 and y = 1 into the equation to solve for y'.
y^2 = 1, 2y = 2, and the y' terms combine as y'*(2x y + 2x) = y'*(2x(y+1)) = y'*(2*3*(1+1)) = y'*12.
Thus the equation becomes 1 + 12 y' + 2 = 0, which simplifies to 3 + 12 y' = 0.
y' = -3/12 = -1/4.
Now evaluate the provided answer options:
- Option 1: −1/4. This matches the computed slope -1/4, so this option is consistent with the derivation.
- Option 2: −1/3. This would imply a different coefficient balance in the y' terms; plugging into the differentiated equation would not satisfy 0 on the right-hand side, so this is incorrect.
- Option 3: 1/3. A positive slope contradicts the algebraic result; substituting would not yield 0, so it's incorrect.
- Option 4: 5/2. This is far from the calculated slope and would not satisfy the implicit differentiation equation at (3,1), so it's incorrect.
In summary, the correct slope is −1/4, corresponding to the first option, while the other options do not satisfy the differentiated equation at the given point.

Use implicit differentiation to find the slope of the tangent line to the curve defined by x y 2 + 2 x y = 5 at the point ( 3 , 1 ) . 单项选择题

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