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

x
y
4
+
6
x
y
=
5
at the point
(
5
,
1
)
.

Question

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

x
y
4
+
6
x
y
=
5
at the point 
(
5
,
1
)
.

BlackTom AI · Accepted Answer

We start by restating the problem and the given data to ensure we’re evaluating the same thing.
- Question: Use implicit differentiation to find the slope of the tangent line to the curve defined by x^4 + 6xy = 5 at the point (5, 1).
- Answer options: 5/6, 1/5, -1/7, -7/50.

Next, we check whether the provided point actually lies on the curve, because a tangent slope at a point on the curve only makes sense if the point is on the curve.
- Evaluate the left-hand side at (x, y) = (5, 1): x^4 + 6xy = 5^4 + 6·5·1 = 625 + 30 = 655.
- Compare to the right-hand side, which is 5. Since 655 ≠ 5, the point (5, 1) is not on the curve defined by x^4 + 6xy = 5.

Because the point does not lie on the curve, there is no tangent line to the curve at that point, and therefore no legitimate slope to compute at (5, 1) for this curve.

If one were to proceed formally with the implicit differentiation step (assuming the point were on the curve), you differentiate implicitly:
- d/dx(x^4) = 4x^3.
- d/dx(6xy) uses the product rule: 6(x dy/dx + y).
- The derivative equation becomes 4x^3 + 6(x dy/dx + y) = 0.
- Solve for dy/dx: 4x^3 + 6y + 6x dy/dx = 0 ⇒ 6x dy/dx = -(4x^3 + 6y) ⇒ dy/dx = -(4x^3 + 6y)/(6x).

If we plug in the coordinates (x, y) = (5, 1) into this derivative (even though the point isn’t on the curve, this shows what the slope would be from the formal formula):
- Numerator: 4x^3 + 6y = 4·125 + 6·1 = 500 + 6 = 506.
- Denominator: 6x = 6·5 = 30.
- dy/dx = -506/30 = -253/15, which is approximately -16.87.

Notice that this value, -253/15, does not match any of the provided answer choices (5/6, 1/5, -1/7, -7/50). The mismatch arises because the premise is inconsistent: the point (5, 1) is not on the curve, so the question as stated cannot yield a valid slope compatible with the given options.

In summary, the key issue is the point’s exclusion from the curve; all listed answer choices fail to reflect the actual situation. If someone still evaluates the derivative at (5, 1) using the standard formula, they would obtain -253/15, which again is not among the choices, reinforcing that the stated problem is inconsistent.

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 4 + 6 x y = 5 at the point ( 5 , 1 ) .

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