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.

MATH1062_MATH1005_MATH1023 MATH1062/1023 Calculus Quiz 8

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 ) .

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