Let x^(4)+y^(2)=17. What is the value of (d^(2)y)/(dx^(2)) at the point (-2,1) ? Give an exact number.

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Let  x^(4)+y^(2)=17. What is the value of  (d^(2)y)/(dx^(2)) at the point  (-2,1) ? Give an exact number.

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Differentiate Equation: Differentiate both sides of the equation x^(4) + y^(2) = 17 with respect to x. Using the power rule for differentiation, the derivative of x^(4) with respect to x is 4x^(3), and the derivative of y^(2) with respect to x is 2y(dy/dx) since y is a function of x. The derivative of a constant is 0. So, the differentiation gives us 4x^(3) + 2y(dy/dx) = 0.Solve for dy/dx: Solve for dy/dx. From the previous step, we have 4x^(3) + 2y(dy/dx) = 0. Rearrange the equation to solve for dy/dx: 2y(dy/dx) = -4x^(3). Now, divide both sides by 2y to isolate dy/dx: (dy/dx) = -4x^(3) / (2y). Simplify the equation: (dy/dx) = -2x^(3) / y.Differentiate dy/dx: Differentiate both sides of the equation (dy/dx) = -2x^(3) / y again with respect to x to find d^(2)y/dx^(2). Using the quotient rule for differentiation, which is (v(u')) - (u(v')) / v^(2), where u = -2x^(3) and v = y, we get: d^(2)y/dx^(2) = [y(-6x^(2)) - (-2x^(3))(dy/dx)] / y^(2).Substitute dy/dx: Substitute the value of dy/dx from Step 2 into the equation from Step 3. We have d^(2)y/dx^(2) = [y(-6x^(2)) - (-2x^(3))(-2x^(3) / y)] / y^(2). Simplify the equation: d^(2)y/dx^(2) = [-6x^(2)y + (4x^(6) / y)] / y^(2).Substitute Point: Substitute the point (-2,1) into the equation from Step 4. We have d^(2)y/dx^(2) = [-6(-2)^(2)(1) + (4(-2)^(6) / 1)] / 1^(2). Calculate the values: d^(2)y/dx^(2) = [-6(4)(1) + (4(64) / 1)] / 1. Simplify the equation: d^(2)y/dx^(2) = [-24 + 256] / 1.Calculate Final Value: Calculate the final value of d^(2)y/dx^(2) at the point (-2,1). We have d^(2)y/dx^(2) = -24 + 256. Simplify the equation: d^(2)y/dx^(2) = 232.

Let $x^{4}+y^{2}=17$ . $\newline$ What is the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(-2,1)$ ? $\newline$ Give an exact number.

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Let $x^{4}+y^{2}=17$ . $\newline$ What is the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(-2,1)$ ? $\newline$ Give an exact number.