Use the curve x^2-x y+y^2=1. Which of the following is equal to (d^2 y)/(d x^2) ? (A) -(6)/((2 y

step 1 Let's use the curve x squared minus xy plus y squared equals 1. So you want to know what the sec

Use the curve x^2-x y+y^2=1. Which of the following is equal...

Use the curve $x^{2}-x y+y^{2}=1$.
Which of the following is equal to $\frac{d^{2} y}{d x^{2}} ?$
(A) $-\frac{6}{(2 y-x)^{3}}$
(B) $\frac{10 y^{2}-10 x^{2}-10 x y}{(2 y-x)^{3}}$
(C) $\frac{8 x^{2}-4 x y+8 y^{2}}{(x-2 y)^{3}}$
(D) $\frac{10 x^{2}+10 y^{2}}{(x-2 y)^{3}}$
(E) $\frac{2}{x}$

Key Concepts

Algebraic Manipulation

Algebraic manipulation plays a key role after differentiating implicitly. It involves factoring, rearranging equations, and combining like terms to simplify the expressions for the derivatives, making them easier to interpret and compare with provided options.

Chain Rule

The chain rule is a fundamental tool in calculus used when differentiating composite functions. In the context of implicitly defined functions, when differentiating terms involving y, the chain rule is applied to account for the inner function y(x), which is critical while finding both the first and second derivatives.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent variable is not isolated explicitly. In problems involving curves defined by equations like x² - xy + y² = 1, this method allows us to compute the derivative dy/dx by differentiating both sides of the equation with respect to x, treating y as a function of x.

Second Derivative

The second derivative, d²y/dx², provides information about the concavity of a curve. After finding the first derivative through implicit differentiation, the process is repeated—typically using rules from calculus—to obtain the second derivative by differentiating dy/dx with respect to x, including accounting for the dependence of y on x.

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