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 Graphs of Functions: 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 = 5x^4... 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 $\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$ in the form $2x^p - x^q$ , first simplify the expression:

Rewrite the denominator:
$\sqrt{x} = x^{1/2}.$
Thus, we have:
$\frac{2x^{-2}-x^{-3}}{x^{1/2}} = 2x^{-2 - \frac{1}{2}} - x^{-3 - \frac{1}{2}}.$
This simplifies to:
$2x^{-\frac{5}{2}} - x^{-\frac{7}{2}}.$
Therefore, we can identify the values as:
- $p = -\frac{5}{2}$
- $q = -\frac{7}{2}$ .

Step 2

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

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Answer

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

Differentiate each term:
- For $5x^4$ $5 x^{4}$ :
  - The derivative is $20x^3.$
- For $-3$ $- 3$ :
  - The derivative is $0.$
- For the term $\frac{2x^{-2}-x^{-3}}{\sqrt{x}}$ $\frac{2 x ^{- 2} - x ^{- 3}}{x}$ , use the quotient rule or simplify first:
  - We already simplified it in part (a) to:
    
    $2x^{-\frac{5}{2}} - x^{-\frac{7}{2}}.$
  - Differentiating gives:
    - For $2x^{-\frac{5}{2}}$ :
      
      $\frac{d}{dx}(2x^{-\frac{5}{2}}) = 2 \cdot -\frac{5}{2}x^{-\frac{7}{2}} = -5x^{-\frac{7}{2}}.$
    - For $-x^{-\frac{7}{2}}$ :
      
      $\frac{d}{dx}(-x^{-\frac{7}{2}}) = -(-\frac{7}{2})x^{-\frac{9}{2}} = \frac{7}{2}x^{-\frac{9}{2}}.$
Combine all derivatives:

$\frac{dy}{dx} = 20x^3 - 5x^{-\frac{7}{2}} + \frac{7}{2}x^{-\frac{9}{2}}.$
For simplification, express in standard form:

$\frac{dy}{dx} = 20x^3 - 5x^{-\frac{7}{2}} + \frac{7}{2}x^{-\frac{9}{2}}.$

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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