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If 4x-4-4x^(2)=y^(2)-y^(3) then find (dy)/(dx) at the point (1,2). Answer: (dy)/(dx)|_((1,2))=

If 4x−4−4x2=y2−y3 4 x-4-4 x^{2}=y^{2}-y^{3} then find dydx \frac{d y}{d x} at the point (1,2) (1,2) .\newlineAnswer: dydx∣(1,2)= \left.\frac{d y}{d x}\right|_{(1,2)}=

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Q. If 4x−4−4x2=y2−y3 4 x-4-4 x^{2}=y^{2}-y^{3} then find dydx \frac{d y}{d x} at the point (1,2) (1,2) .\newlineAnswer: dydx∣(1,2)= \left.\frac{d y}{d x}\right|_{(1,2)}=
  1. Differentiate and Simplify: First, we need to differentiate both sides of the equation with respect to xx to find dydx\frac{dy}{dx}. The equation is 4x−4−4x2=y2−y34x - 4 - 4x^2 = y^2 - y^3. Differentiating the left side with respect to xx gives us the derivative of 4x4x, which is 44, minus the derivative of 44, which is 00, minus the derivative of 4x24x^2, which is 8x8x. Differentiating the right side with respect to xx is a bit more complex because it involves dydx\frac{dy}{dx}11, which is a function of xx. We need to use the chain rule. The derivative of dydx\frac{dy}{dx}33 with respect to xx is dydx\frac{dy}{dx}55, and the derivative of dydx\frac{dy}{dx}66 with respect to xx is dydx\frac{dy}{dx}88. So, the differentiated equation is dydx\frac{dy}{dx}99.
  2. Solve for dydx\frac{dy}{dx}: Now we need to solve for dydx\frac{dy}{dx}. We can factor out dydx\frac{dy}{dx} on the right side of the equation to get dydx(2y−3y2)\frac{dy}{dx}(2y - 3y^2). So, we have 4−8x=dydx(2y−3y2)4 - 8x = \frac{dy}{dx}(2y - 3y^2). To solve for dydx\frac{dy}{dx}, we divide both sides by (2y−3y2)(2y - 3y^2). This gives us dydx=4−8x2y−3y2\frac{dy}{dx} = \frac{4 - 8x}{2y - 3y^2}.
  3. Substitute Point and Calculate: Next, we need to substitute the point (1,2)(1,2) into the equation to find the value of dydx\frac{dy}{dx} at that point.\newlineSubstituting x=1x = 1 and y=2y = 2 into the equation dydx=(4−8x)(2y−3y2)\frac{dy}{dx} = \frac{(4 - 8x)}{(2y - 3y^2)} gives us dydx=(4−8⋅1)(2⋅2−3⋅22)\frac{dy}{dx} = \frac{(4 - 8\cdot 1)}{(2\cdot 2 - 3\cdot 2^2)}.
  4. Perform Calculations: Now we perform the calculations.\newlineCalculating the numerator: 4−8×1=4−8=−44 - 8 \times 1 = 4 - 8 = -4.\newlineCalculating the denominator: 2×2−3×22=4−3×4=4−12=−82 \times 2 - 3 \times 2^2 = 4 - 3 \times 4 = 4 - 12 = -8.\newlineSo, dydx=−4−8\frac{dy}{dx} = \frac{-4}{-8}.
  5. Simplify Fraction: Finally, we simplify the fraction −48-\frac{4}{8} to get the value of dydx\frac{dy}{dx}. Simplifying the fraction gives us dydx=12\frac{dy}{dx} = \frac{1}{2}. Therefore, the value of dydx\frac{dy}{dx} at the point (1,2)(1,2) is 12\frac{1}{2}.

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