Question

If e²y - eʸ²-y) = x⁴ - x², then the value of dy/dx at the point (1,0) is

a) 1
b) -1
c) 0
d) 2

Answer 1

The value of dy/dx at the point (1,0) is found by implicit differentiation of the given function and solving for dy/dx, which turns out to be 2/3, not matching any of the given options a through d.

Step-by-step explanation:

The student is asking how to find the value of dy/dx at the point (1,0) given the function e²y - eⁿ(y²-y) = x⁴ - x². To solve this, we need to implicitly differentiate both sides of the equation with respect to x.

Implicit differentiation of e²y gives us 2e²y(dy/dx) and for eⁿ(y²-y) gives us eⁿ(y²-y)(2y-1)(dy/dx) after applying the chain rule. For the right side, differentiation of x⁴ - x² yields 4x³ - 2x.

After differentiating, we get 2e²y(dy/dx) - eⁿ(y²-y)(2y-1)(dy/dx) = 4x³ - 2x. At the point (1,0), e²y and eⁿ(y²-y) equals 1 and (2y-1) equals -1, simplifying the left side of the equation to 2(dy/dx) + (dy/dx). The right side at the point (1,0) simplifies to 2. Solving for dy/dx, we get 3(dy/dx) = 2, thus dy/dx = 2/3.

Therefore, the value of dy/dx at the point (1,0) is not one of the provided options (a, b, c, or d). Instead, the correct answer is 2/3.