Let y^(4)+5x=11. What is the value of (d^(2)y)/(dx^(2)) at the point (2,1) ? Give an exact number.

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Let  y^(4)+5x=11. 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: Given the equation y^(4) + 5x = 11, we need to find the second derivative of y with respect to x, (d^(2)y)/(dx^(2)), at the point (2,1). To do this, we first need to differentiate both sides of the equation with respect to x to find the first derivative (dy/dx). Differentiate y^(4) + 5x = 11 with respect to x: d/dx(y^(4)) + d/dx(5x) = d/dx(11) Using the power rule and the constant rule for differentiation, we get: 4y^(3) * (dy/dx) + 5 = 0Solve for dy/dx: Now we need to solve for dy/dx: 4y^(3) * (dy/dx) + 5 = 0 Subtract 5 from both sides: 4y^(3) * (dy/dx) = -5 Divide both sides by 4y^(3): (dy/dx) = -5 / (4y^(3))Find Second Derivative: Next, we need to find the second derivative (d^(2)y)/(dx^(2)). To do this, we differentiate (dy/dx) with respect to x again. However, since dy/dx is a function of y, and y is a function of x, we need to use the chain rule for differentiation. Differentiate (dy/dx) = -5 / (4y^(3)) with respect to x: (d^(2)y)/(dx^(2)) = d/dx(-5 / (4y^(3))) Using the chain rule and the power rule, we get: (d^(2)y)/(dx^(2)) = -5 * d/dx(1 / (4y^(3))) (d^(2)y)/(dx^(2)) = -5 * (-3) * (1 / (4y^(4))) * (dy/dx) (d^(2)y)/(dx^(2)) = 15 * (1 / (4y^(4))) * (dy/dx)Substitute dy/dx: Now we need to substitute the expression we found for dy/dx into the equation for the second derivative: (d^(2)y)/(dx^(2)) = 15 * (1 / (4y^(4))) * (-5 / (4y^(3))) Simplify the expression: (d^(2)y)/(dx^(2)) = 15 * (-5) / (16y^(7)) (d^(2)y)/(dx^(2)) = -75 / (16y^(7))Evaluate at (2,1): Finally, we need to evaluate the second derivative at the point (2,1). Since y = 1 at this point, we substitute y = 1 into the expression for the second derivative: (d^(2)y)/(dx^(2)) at (2,1) = -75 / (16 * (1)^(7)) Simplify the expression: (d^(2)y)/(dx^(2)) at (2,1) = -75 / 16

Let $y^{4}+5 x=11$ . $\newline$ 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 $y^{4}+5 x=11$ . $\newline$ What is the value of $\frac{d^{2} y}{d x^{2}}$ at the point $(2,1)$ ? $\newline$ Give an exact number.