Given that $$\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$$ can be written in the form $2x^p - x^q$, (a) write down the value of $p$ and the value of $q$. Given that $y = 5x^4 - 3 + \frac{2x^{-2}-x^{-3}}{\sqrt{x}}$, (b) find $\frac{dy}{dx}$, simplifying the coefficient of each term.

A-Level Edexcel Maths Pure Polynomials: Given that $$\frac{2x^{-2}-x^{

Given that $$\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$$ can be written in the form $2x^p - x^q$, (a) write down the value of $p$ and the value of $q$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 1

Question 8

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Given that $$\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$$ can be written in the form $2x^p - x^q$, (a) write down the value of $p$ and the value of $q$. Given that $y... show full transcript

Worked Solution & Example Answer:Given that $$\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$$ can be written in the form $2x^p - x^q$, (a) write down the value of $p$ and the value of $q$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 1

Step 1

write down the value of p and the value of q.

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Answer

To express the given function in the form $2x^p - x^q$ , we first simplify the original expression:

$\frac{2x^{-2}-x^{-3}}{\sqrt{x}} = \frac{2x^{-2}-x^{-3}}{x^{1/2}}$

Combining the exponents:

$= 2x^{-2-1/2} - x^{-3-1/2} = 2x^{-2.5} - x^{-3.5}$

Thus, we can identify:

$p = -2.5$
$q = -3.5$ .

Step 2

find $\frac{dy}{dx}$, simplifying the coefficient of each term.

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104 rated

Answer

Starting from the equation:

$y = 5x^4 - 3 + \frac{2x^{-2}-x^{-3}}{\sqrt{x}}$

We differentiate term by term:

For $5x^4$ , the derivative is $\frac{dy}{dx} = 20x^3$ .
The derivative of $-3$ is $0$ .
For the term $\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$ , rewrite it as:

$2x^{-2} \cdot x^{-1/2} - x^{-3} \cdot x^{-1/2} = 2x^{-2.5} - x^{-3.5}$

Then differentiate:

$\frac{d}{dx}(2x^{-2.5}) = -5x^{-3.5} \quad \text{and} \quad \frac{d}{dx}(-x^{-3.5}) = 3.5x^{-4.5}$

Now combining all derivatives gives:

$\frac{dy}{dx} = 20x^3 - 5x^{-3.5} + 3.5x^{-4.5}$

Simplifying the expression:

$\frac{dy}{dx} = 20x^3 - \frac{5}{x^{3.5}} + \frac{3.5}{x^{4.5}}$

Given that $$\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$$ can be written in the form $2x^p - x^q$, (a) write down the value of $p$ and the value of $q$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 1

Worked Solution & Example Answer:Given that $$\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$$ can be written in the form $2x^p - x^q$, (a) write down the value of $p$ and the value of $q$ - Edexcel - A-Level Maths Pure - Question 8 - 2009 - Paper 1

write down the value of p and the value of q.

find \(\frac{dy}{dx}\), simplifying the coefficient of each term.

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