A-Level Edexcel Maths Pure Vectors in 3 Dimensions: The curve C has equation $$3^{x

The curve C has equation $$3^{x} + xy - y^{2} + 5 = 0$$ Show that $rac{dy}{dx}$ at the point (1, 3) on the curve C can be written in the form $rac{1}{ ho} ext{ln}( u e^{3})$, where $ ho$ and $ u$ are integers to be found. - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

Question 6

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The curve C has equation $$3^{x} + xy - y^{2} + 5 = 0$$ Show that $rac{dy}{dx}$ at the point (1, 3) on the curve C can be written in the form $rac{1}{ ho} ext{l... show full transcript

Worked Solution & Example Answer:The curve C has equation $$3^{x} + xy - y^{2} + 5 = 0$$ Show that $rac{dy}{dx}$ at the point (1, 3) on the curve C can be written in the form $rac{1}{ ho} ext{ln}( u e^{3})$, where $ ho$ and $ u$ are integers to be found. - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

Step 1

Differentiate implicitly

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Answer

To find $rac{dy}{dx}$ , we differentiate the given equation implicitly:

$rac{d}{dx}(3^{x}) + rac{d}{dx}(xy) - rac{d}{dx}(y^{2}) + rac{d}{dx}(5) = 0$

Using chain rule, the derivatives are:

For $3^{x}$ , the derivative is $3^{x} ext{ln}(3)$ .
For $xy$ , we use the product rule: $x rac{dy}{dx} + y$ .
For $y^{2}$ , the derivative is $2y rac{dy}{dx}$ .

Thus, the differentiated equation becomes:

$3^{x} ext{ln}(3) + rac{dy}{dx}(x + 2y) = 0$

Step 2

Evaluate at the point (1, 3)

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Answer

Substituting $(x, y) = (1, 3)$ into the differentiated equation:

$3^{1} ext{ln}(3) + rac{dy}{dx}(1 + 2(3)) = 0$

This simplifies to:

$3 ext{ln}(3) + rac{dy}{dx}(7) = 0$

Solving for $rac{dy}{dx}$ gives:

$rac{dy}{dx} = - rac{3 ext{ln}(3)}{7}$

Step 3

Express in the required form

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101 rated

Answer

We need to express $rac{dy}{dx}$ in the form $rac{1}{ ho} ext{ln}( u e^{3})$ .

We can rewrite our expression as:

$- rac{3 ext{ln}(3)}{7} = - rac{1}{7} ext{ln}(3^{3}) \ = - rac{1}{7} ext{ln}(3 imes e^{3})$

Thus, we identify:

$ho = 7$
$u = 3$

The curve C has equation $$3^{x} + xy - y^{2} + 5 = 0$$ Show that $ rac{dy}{dx}$ at the point (1, 3) on the curve C can be written in the form $ rac{1}{ ho} ext{ln}( u e^{3})$, where $ ho$ and $ u$ are integers to be found. - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1

Differentiate implicitly

Evaluate at the point (1, 3)

Express in the required form

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The curve C has equation $$3^{x} + xy - y^{2} + 5 = 0$$ Show that $rac{dy}{dx}$ at the point (1, 3) on the curve C can be written in the form $rac{1}{ ho} ext{ln}( u e^{3})$, where $ ho$ and $ u$ are integers to be found. - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 1