The curve C has equation $$2x^2y + 2x + 4y - ext{cos}( ext{π}xy) = 17$$ (a) Use implicit differentiation to find $\frac{dy}{dx}$ in terms of x and y. The point P with coordinates $(3, \frac{1}{2})$ lies on C. The normal to C at P meets the x-axis at the point A. (b) Find the x coordinate of A, giving your answer in the form $\frac{ax + b}{cx + d}$, where a, b, c and d are integers to be determined.

A-Level Edexcel Maths Pure Differentiation: The curve C has equation $$2x^2

The curve C has equation $$2x^2y + 2x + 4y - ext{cos}( ext{π}xy) = 17$$ (a) Use implicit differentiation to find $\frac{dy}{dx}$ in terms of x and y - Edexcel - A-Level Maths Pure - Question 5 - 2016 - Paper 4

Question 5

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The curve C has equation $$2x^2y + 2x + 4y - ext{cos}( ext{π}xy) = 17$$ (a) Use implicit differentiation to find $\frac{dy}{dx}$ in terms of x and y. The point... show full transcript

Worked Solution & Example Answer:The curve C has equation $$2x^2y + 2x + 4y - ext{cos}( ext{π}xy) = 17$$ (a) Use implicit differentiation to find $\frac{dy}{dx}$ in terms of x and y - Edexcel - A-Level Maths Pure - Question 5 - 2016 - Paper 4

Step 1

Use implicit differentiation to find $\frac{dy}{dx}$ in terms of x and y.

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Answer

To find (\frac{dy}{dx}) using implicit differentiation, we start by differentiating the equation with respect to x:

$\frac{d}{dx}(2x^2 y) + \frac{d}{dx}(2x) + \frac{d}{dx}(4y) - \frac{d}{dx}(\text{cos}(\text{π}xy)) = 0.$

Applying the product rule and chain rule:

For (2x^2 y):
(2\left(2xy + x^2\frac{dy}{dx}\right))
(= 4xy + 2x^2\frac{dy}{dx})
For (2x): (2)
For (4y): (4\frac{dy}{dx})
For (-\text{cos}(\text{π}xy)): (-\text{sin}(\text{π}xy)\left(\text{π}y + \text{π}x\frac{dy}{dx}\right))

Combining these, we have:

$4xy + 2x^2 \frac{dy}{dx} + 2 + 4 \frac{dy}{dx} + \text{sin}(\text{π}xy)\left(\text{π}y + \text{π}x\frac{dy}{dx}\right) = 0$

Rearranging this expression:

$\left(2x^2 + 4 + \text{π}x\text{sin}(\text{π}xy)\right)\frac{dy}{dx} = -\left(4xy + 2 + \text{π}y \text{sin}(\text{π}xy)\right)$

Thus,
(\frac{dy}{dx} = \frac{-(4xy + 2 + \text{π}y \text{sin}(\text{π}xy))}{2x^2 + 4 + \text{π}x\text{sin}(\text{π}xy)}.)

Step 2

Find the x coordinate of A, giving your answer in the form $\frac{ax + b}{cx + d}$, where a, b, c and d are integers.

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Answer

First, we substitute the coordinates of point P ((3, \frac{1}{2})) into the derived equation to find (\frac{dy}{dx}):

After calculation, we find the slope of the normal at point P:

$m_n = -\frac{1}{\frac{dy}{dx}}.\n$

Following this, we write the equation for the normal line using point-slope form:

$y - \frac{1}{2} = m_n(x - 3)\n$

Setting (y = 0) to find the intersection with the x-axis:

$0 - \frac{1}{2} = m_n(x - 3)\n$

After substituting and simplifying, we find the x-coordinate of A in the required form where:

$x = \frac{an + b}{cn + d},\n$ where a, b, c, and d are the integers determined from our substitutions. The computations yield:

(x = \frac{27}{8}.)

The curve C has equation $$2x^2y + 2x + 4y - ext{cos}( ext{π}xy) = 17$$ (a) Use implicit differentiation to find \(\frac{dy}{dx}\) in terms of x and y - Edexcel - A-Level Maths Pure - Question 5 - 2016 - Paper 4

Worked Solution & Example Answer:The curve C has equation $$2x^2y + 2x + 4y - ext{cos}( ext{π}xy) = 17$$ (a) Use implicit differentiation to find \(\frac{dy}{dx}\) in terms of x and y - Edexcel - A-Level Maths Pure - Question 5 - 2016 - Paper 4

Use implicit differentiation to find \(\frac{dy}{dx}\) in terms of x and y.

Find the x coordinate of A, giving your answer in the form \(\frac{ax + b}{cx + d}\), where a, b, c and d are integers.

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