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If then find at the point .Answer:
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Q. If then find at the point .Answer:
- Apply Implicit Differentiation: To find the derivative of with respect to , , for the given equation, we need to use implicit differentiation because is not isolated on one side of the equation.
- Differentiate Each Term: Differentiate both sides of the equation with respect to , applying the chain rule where necessary. The equation is .Differentiating term by term, we get:
- Solve for (\frac{dy}{dx}): Differentiate each term:\(\newline\$\frac{d}{dx}(-2x^2) = -4x\)\(\newline\)\(\frac{d}{dx}(y^2) = 2y \cdot (\frac{dy}{dx})\) (using the chain rule)\(\newline\)\(\frac{d}{dx}(y^3) = 3y^2 \cdot (\frac{dy}{dx})\) (using the chain rule)\(\newline\)\(\frac{d}{dx}(3x) = 3\)\(\newline\)\(\frac{d}{dx}(0) = 0\)\(\newline\)So, we have \(-4x + 2y(\frac{dy}{dx}) + 3y^2(\frac{dy}{dx}) + 3 = 0\)
- Factor Out \(\frac{dy}{dx}\): Now, we need to solve for \(\frac{dy}{dx}\). Group all the terms involving \(\frac{dy}{dx}\) on one side and the rest on the other side:\(\newline\)\(2y\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 4x - 3\)
- Isolate \(\frac{dy}{dx}\): Factor out \(\frac{dy}{dx}\) from the left side:\(\newline\)\(\frac{dy}{dx}(2y + 3y^2) = 4x - 3\)
- Substitute Values: Divide both sides by \((2y + 3y^2)\) to isolate \((\frac{dy}{dx}):\)\[(\frac{dy}{dx}) = \frac{(4x - 3)}{(2y + 3y^2)}\]
- Calculate \((\frac{dy}{dx}): Now we need to substitute \$x = 2\) and \(y = 1\) into the equation to find the value of \((\frac{dy}{dx})\) at the point \((2,1)\):\(\newline\)\((\frac{dy}{dx})|_{(\(2\),\(1\))} = \frac{(\(4\)(\(2\)) - \(3\))}{(\(2\)(\(1\)) + \(3\)(\(1\))^\(2\))}
- Calculate \((\frac{dy}{dx}): Now we need to substitute \$x = 2\) and \(y = 1\) into the equation to find the value of \((\frac{dy}{dx})\) at the point \((2,1)\):
\((\frac{dy}{dx})|_{(2,1)} = \frac{(4(2) - 3)}{(2(1) + 3(1)^2)}\)Calculate the value:
\((\frac{dy}{dx})|_{(2,1)} = \frac{(8 - 3)}{(2 + 3)}\)
\((\frac{dy}{dx})|_{(2,1)} = \frac{5}{5}\)
\((\frac{dy}{dx})|_{(2,1)} = 1\)
