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

Question

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

Bytelearn · Accepted Answer

Given Equation Differentiation: We are given the equation y^(4) - 2x = 5. To find the second derivative of y with respect to x, we first need to differentiate both sides of the equation with respect to x. Differentiate y^(4) - 2x = 5 with respect to x: d/dx(y^(4)) - d/dx(2x) = d/dx(5) 4y^(3) * (dy/dx) - 2 = 0First Derivative Calculation: Now, we need to solve for dy/dx, which is the first derivative of y with respect to x. 4y^(3) * (dy/dx) = 2 (dy/dx) = 2 / (4y^(3)) (dy/dx) = 1 / (2y^(3))Second Derivative Calculation: Next, we differentiate the first derivative with respect to x to find the second derivative. d/dx(dy/dx) = d/dx(1 / (2y^(3))) Using the chain rule and the quotient rule, we get: (d^(2)y)/(dx^(2)) = -6y^(2) * (dy/dx)^(2) / (2y^(3))Substitution for Given Point: We substitute the value of dy/dx from the previous step into the expression for the second derivative. (d^(2)y)/(dx^(2)) = -6y^(2) * (1 / (2y^(3)))^(2) / (2y^(3)) (d^(2)y)/(dx^(2)) = -6y^(2) * (1 / (4y^(6))) / (2y^(3)) (d^(2)y)/(dx^(2)) = -6y^(2) / (8y^(9)) (d^(2)y)/(dx^(2)) = -3 / (4y^(7))Final Result: Finally, we substitute the given point (-2,1) into the expression for the second derivative. (d^(2)y)/(dx^(2)) at the point (-2,1) is: (d^(2)y)/(dx^(2)) = -3 / (4 * 1^(7)) (d^(2)y)/(dx^(2)) = -3 / 4

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

Full solution

More problems from Evaluate expression when two complex numbers are given

Related topics

Let y4−2x=5 y^{4}-2 x=5 y4−2x=5.\newlineWhat is the value of d2ydx2 \frac{d^{2} y}{d x^{2}} dx2d2y​ at the point (−2,1) (-2,1) (−2,1) ?\newlineGive an exact number.

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