Let x^(3)+y^(2)=24. What is the value of (d^(2)y)/(dx^(2)) at the point (2,4) ? Give an exact number.

Question

Let  x^(3)+y^(2)=24. What is the value of  (d^(2)y)/(dx^(2)) at the point  (2,4) ? Give an exact number.

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Implicit Differentiation: Given the equation x^3 + y^2 = 24, we need to find the second derivative of y with respect to x at the point (2,4). To do this, we first need to find the first derivative dy/dx by implicit differentiation. Differentiate both sides of the equation with respect to x: d/dx (x^3) + d/dx (y^2) = d/dx (24) 3x^2 + 2y(dy/dx) = 0 Now, solve for dy/dx: 2y(dy/dx) = -3x^2 (dy/dx) = -3x^2 / (2y)Second Derivative Calculation: Next, we need to find the second derivative d^2y/dx^2. To do this, we differentiate dy/dx with respect to x again, using implicit differentiation and the chain rule. Differentiate -3x^2 / (2y) with respect to x: d/dx (dy/dx) = d/dx (-3x^2 / (2y)) Using the quotient rule, we get: d^2y/dx^2 = (2y * d/dx (-3x^2) - (-3x^2) * d/dx (2y)) / (2y)^2 d^2y/dx^2 = (2y * (-6x) - (-3x^2) * 2(dy/dx)) / (2y)^2Substitution of Values: Now we need to substitute dy/dx from the first derivative we found earlier into the second derivative equation. Substitute dy/dx = -3x^2 / (2y) into the second derivative: d^2y/dx^2 = (2y * (-6x) - (-3x^2) * 2(-3x^2 / (2y))) / (2y)^2 Simplify the equation: d^2y/dx^2 = (-12xy + 9x^4 / y) / (4y^2)Final Result: Finally, we substitute the point (2,4) into the second derivative to find the value of d^2y/dx^2 at that point. Substitute x = 2 and y = 4 into the second derivative: d^2y/dx^2 = (-12*2*4 + 9*2^4 / 4) / (4*4^2) Simplify the equation: d^2y/dx^2 = (-96 + 144 / 4) / 64 d^2y/dx^2 = (-96 + 36) / 64 d^2y/dx^2 = -60 / 64 Reduce the fraction: d^2y/dx^2 = -15 / 16

Let $x^{3}+y^{2}=24$ . $\newline$ What is the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(2,4)$ ? $\newline$ Give an exact number.

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Let $x^{3}+y^{2}=24$ . $\newline$ What is the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(2,4)$ ? $\newline$ Give an exact number.