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Let x^(2)+y^(2)=25. What is the value of (d^(2)y)/(dx^(2)) at the point (4,3) ? Give an exact number.

Let x2+y2=25 x^{2}+y^{2}=25 .\newlineWhat is the value of d2ydx2 \frac{d^{2} y}{d x^{2}} at the point (4,3) (4,3) ?\newlineGive an exact number.

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Q. Let x2+y2=25 x^{2}+y^{2}=25 .\newlineWhat is the value of d2ydx2 \frac{d^{2} y}{d x^{2}} at the point (4,3) (4,3) ?\newlineGive an exact number.
  1. Implicit Differentiation: We are given the equation of a circle x2+y2=25x^2 + y^2 = 25. To find the value of d2ydx2\frac{d^2y}{dx^2} at the point (4,3)(4,3), we first need to implicitly differentiate the equation with respect to xx to find dydx\frac{dy}{dx}.\newlineDifferentiating both sides of the equation with respect to xx, we get:\newline2x+2ydydx=02x + 2y\frac{dy}{dx} = 0
  2. Finding dydx\frac{dy}{dx}: Now we solve for dydx\frac{dy}{dx}:
    2y(dydx)=−2x2y\left(\frac{dy}{dx}\right) = -2x
    dydx=−2x2y\frac{dy}{dx} = \frac{-2x}{2y}
    dydx=−xy\frac{dy}{dx} = \frac{-x}{y}
  3. Substitute Point: We substitute the point (4,3)(4,3) into the equation for dydx\frac{dy}{dx} to find its value at that point:\newline(dydx)(\frac{dy}{dx}) at (4,3)=−43(4,3) = -\frac{4}{3}
  4. Second Derivative: Next, we need to differentiate dydx\frac{dy}{dx} with respect to xx again to find d2ydx2\frac{d^2y}{dx^2}. This is the second derivative we are looking for.\newlineDifferentiating −xy-\frac{x}{y} with respect to xx, we get:\newlined2ydx2=ddx(−xy)\frac{d^2y}{dx^2} = \frac{d}{dx} \left(-\frac{x}{y}\right)
  5. Quotient Rule: To differentiate −x/y-x/y, we will use the quotient rule: (vdu−udv)/v2(vdu - udv) / v^2, where u=−xu = -x and v=yv = y. Let's find dudx\frac{du}{dx} and dvdx\frac{dv}{dx}: dudx=ddx(−x)=−1\frac{du}{dx} = \frac{d}{dx} (-x) = -1 dvdx=ddx(y)=dydx\frac{dv}{dx} = \frac{d}{dx} (y) = \frac{dy}{dx} (which we found earlier to be −x/y-x/y)
  6. Simplify Expression: Now we apply the quotient rule:\newlined2ydx2=y(−1)−(−x)(−xy)y2\frac{d^2y}{dx^2} = \frac{y(-1) - (-x)(-\frac{x}{y})}{y^2}\newlined2ydx2=−y−x2yy2\frac{d^2y}{dx^2} = \frac{-y - \frac{x^2}{y}}{y^2}
  7. Substitute Point: We simplify the expression: d2ydx2=−y2−x2y3\frac{d^2y}{dx^2} = \frac{-y^2 - x^2}{y^3}
  8. Substitute Point: We simplify the expression:\newlined2y/dx2=(−y2−x2)/y3d^2y/dx^2 = (-y^2 - x^2) / y^3Now we substitute the point (4,3)(4,3) into the equation for d2y/dx2d^2y/dx^2 to find its value at that point:\newlined2y/dx2d^2y/dx^2 at (4,3)(4,3) = (−(3)2−(4)2)/(3)3(-(3)^2 - (4)^2) / (3)^3\newlined2y/dx2d^2y/dx^2 at (4,3)(4,3) = (−9−16)/27(-9 - 16) / 27\newlined2y/dx2d^2y/dx^2 at (4,3)(4,3) = (4,3)(4,3)11

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