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If 4x+y^(2)+2-y^(3)=-2x^(2) then find (dy)/(dx) at the point (-4,3) Answer: (dy)/(dx)|_((-4,3))=

If 4x+y2+2−y3=−2x2 4 x+y^{2}+2-y^{3}=-2 x^{2} then find dydx \frac{d y}{d x} at the point (−4,3) (-4,3) \newlineAnswer: dydx∣(−4,3)= \left.\frac{d y}{d x}\right|_{(-4,3)}=

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Q. If 4x+y2+2−y3=−2x2 4 x+y^{2}+2-y^{3}=-2 x^{2} then find dydx \frac{d y}{d x} at the point (−4,3) (-4,3) \newlineAnswer: dydx∣(−4,3)= \left.\frac{d y}{d x}\right|_{(-4,3)}=
  1. Differentiate Equation: First, we need to find the derivative of the given equation with respect to xx to find dydx\frac{dy}{dx}. The given equation is 4x+y2+2−y3=−2x24x + y^2 + 2 - y^3 = -2x^2. We will differentiate each term with respect to xx, treating yy as a function of xx (implicit differentiation).
  2. Differentiate 4x4x: Differentiate 4x4x with respect to xx. The derivative of 4x4x with respect to xx is 44.
  3. Differentiate y2y^2: Differentiate y2y^2 with respect to xx. Since yy is a function of xx, we use the chain rule: the derivative of y2y^2 with respect to xx is 2y(dydx)2y(\frac{dy}{dx}).
  4. Differentiate 22: Differentiate 22 with respect to xx. The derivative of a constant is 00.
  5. Differentiate −y3-y^3: Differentiate −y3-y^3 with respect to xx. Again using the chain rule, the derivative of −y3-y^3 with respect to xx is −3y2dydx-3y^2\frac{dy}{dx}.
  6. Differentiate −2x2-2x^2: Differentiate −2x2-2x^2 with respect to xx. The derivative of −2x2-2x^2 with respect to xx is −4x-4x.
  7. Combine Differentiated Terms: Combine all the differentiated terms to form the derivative of the entire equation: 4+2ydydx−3y2dydx=−4x4 + 2y\frac{dy}{dx} - 3y^2\frac{dy}{dx} = -4x.
  8. Isolate dydx\frac{dy}{dx}: Now we need to solve for dydx\frac{dy}{dx}. We can rearrange the terms to isolate dydx\frac{dy}{dx} on one side of the equation: 2ydydx−3y2dydx=−4x−42y\frac{dy}{dx} - 3y^2\frac{dy}{dx} = -4x - 4.
  9. Factor Out (dydx):(\frac{dy}{dx}): Factor out (dydx)(\frac{dy}{dx}) from the left side of the equation: (dydx)(2y−3y2)=−4x−4(\frac{dy}{dx})(2y - 3y^2) = -4x - 4.
  10. Solve for (dydx):(\frac{dy}{dx}): Divide both sides by (2y−3y2)(2y - 3y^2) to solve for (dydx):(\frac{dy}{dx}): (dydx)=−4x−42y−3y2.(\frac{dy}{dx}) = \frac{-4x - 4}{2y - 3y^2}.
  11. Substitute Point: Now we substitute the point (−4,3)(-4,3) into the equation to find the value of (dydx)(\frac{dy}{dx}) at that point: (dydx)∣(−4,3)=−4(−4)−42(3)−3(3)2.(\frac{dy}{dx})|_{(-4,3)} = \frac{-4(-4) - 4}{2(3) - 3(3)^2}.
  12. Calculate Numerator: Calculate the numerator: −4(−4)−4=16−4=12-4(-4) - 4 = 16 - 4 = 12.
  13. Calculate Denominator: Calculate the denominator: 2(3)−3(3)2=6−3(9)=6−27=−212(3) - 3(3)^2 = 6 - 3(9) = 6 - 27 = -21.
  14. Divide Numerator by Denominator: Now, divide the numerator by the denominator to find (dy/dx)(dy/dx) at the point (−4,3)(-4,3): (dy/dx)∣(−4,3)=12−21(dy/dx)|_{(-4,3)} = \frac{12}{-21}.
  15. Simplify Fraction: Simplify the fraction 12−21\frac{12}{-21} to get the final value of (dy/dx)(dy/dx) at the point (−4,3)(-4,3): (dy/dx)∣(−4,3)=−47(dy/dx)|_{(-4,3)} = \frac{-4}{7}.

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