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

Bytelearn · Accepted Answer

Implicit Differentiation: We are given the equation y^(4) - 2x = 5. To find (d^(2)y)/(dx^(2)), we first need to implicitly differentiate the equation with respect to x to find (dy)/(dx). Implicit differentiation of y^(4) - 2x = 5 gives us: 4y^(3)(dy)/(dx) - 2 = 0. Now, solve for (dy)/(dx): 4y^(3)(dy)/(dx) = 2, (dy)/(dx) = 2 / (4y^(3)), (dy)/(dx) = 1 / (2y^(3)).Finding (dy)/(dx): Next, we need to differentiate (dy)/(dx) with respect to x again to find (d^(2)y)/(dx^(2)). This requires using the chain rule since y is a function of x. (d^(2)y)/(dx^(2)) = d/dx[1 / (2y^(3))]. To differentiate this, we use the quotient rule: (d^(2)y)/(dx^(2)) = -3 * (1/2) * y^(-4) * (dy)/(dx).Using Chain Rule: Now we need to substitute the value of (dy)/(dx) that we found in the first step into the expression for (d^(2)y)/(dx^(2)). (d^(2)y)/(dx^(2)) = -3 * (1/2) * y^(-4) * (1 / (2y^(3))), (d^(2)y)/(dx^(2)) = -3 * (1/2) * y^(-4) * (1 / (2y^(3))), (d^(2)y)/(dx^(2)) = -3 / (4y^(7)).Substitute and Evaluate: We are given the point (-2,1) to evaluate (d^(2)y)/(dx^(2)). We substitute y = 1 into the expression for (d^(2)y)/(dx^(2)). (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.

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

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