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Given the equation: $$x^2 + y^2 + 10x + 2y - 4xy = 10$$ (a) Find \( \frac{dy}{dx} \) in terms of x and y, fully simplifying your answer - Edexcel - A-Level Maths Pure - Question 4 - 2014 - Paper 8
Question 4
Given the equation: $$x^2 + y^2 + 10x + 2y - 4xy = 10$$ (a) Find \( \frac{dy}{dx} \) in terms of x and y, fully simplifying your answer. (b) Find the values of y ... show full transcript
Worked Solution & Example Answer:Given the equation: $$x^2 + y^2 + 10x + 2y - 4xy = 10$$ (a) Find \( \frac{dy}{dx} \) in terms of x and y, fully simplifying your answer - Edexcel - A-Level Maths Pure - Question 4 - 2014 - Paper 8
Step 1
Find \( \frac{dy}{dx} \) in terms of x and y
Answer
To find ( \frac{dy}{dx} ), we will use implicit differentiation on the given equation:
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Differentiate both sides with respect to x: [\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) + \frac{d}{dx}(10x) + \frac{d}{dx}(2y) - \frac{d}{dx}(4xy) = \frac{d}{dx}(10)]
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Applying the differentiation rules: [2x + 2y\frac{dy}{dx} + 10 + 2\frac{dy}{dx} - (4y + 4x\frac{dy}{dx}) = 0]
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Rearranging the equation gives: [2x + 10 + 2\frac{dy}{dx} - 4y - 4x\frac{dy}{dx} = 0]
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Combine the ( \frac{dy}{dx} ) terms: [(2 - 4x)\frac{dy}{dx} = 4y - 2x - 10]
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Solving for ( \frac{dy}{dx} ): [\frac{dy}{dx} = \frac{4y - 2x - 10}{2 - 4x}]
Thus, the simplified expression for ( \frac{dy}{dx} ) is: [\frac{dy}{dx} = \frac{2(2y - x - 5)}{2 - 4x}]
Step 2
Find the values of y for which \( \frac{dy}{dx} = 0 \)
Answer
To find the values of y where ( \frac{dy}{dx} = 0 ):
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Set the numerator of ( \frac{dy}{dx} ) equal to zero: [4y - 2x - 10 = 0]
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Rearranging gives: [4y = 2x + 10] [y = \frac{1}{2}(x + 5)]
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Therefore, for ( \frac{dy}{dx} = 0 ), the value of y is: [y = \frac{x + 5}{2}]