A-Level Edexcel Maths Pure Rational Expressions: Given that $y = x^4 + 6x^{-ra

Given that $y = x^4 + 6x^{-rac{1}{2}}$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \: dx$ - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 1

Question 2

$Given-that---$y-=-x^4-+-6x^{--rac{1}{2}}$,-find-in-their-simplest-form---(a)-$\frac{dy}{dx}$---(b)-$\int-y-\:-dx$-Edexcel-A-Level Maths Pure-Question 2-2012-Paper 1.png$

Given that $y = x^4 + 6x^{-rac{1}{2}}$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \: dx$

Worked Solution & Example Answer:Given that $y = x^4 + 6x^{-rac{1}{2}}$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \: dx$ - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 1

Step 1

(a) $\frac{dy}{dx}$

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Answer

To find $rac{dy}{dx}$ , we will differentiate the function:

Use the power rule for differentiation, which states that

$\frac{d}{dx}(x^n) = nx^{n-1}$
Differentiate each term in the expression:
- For the first term $x^4$ :
  
  $\frac{d}{dx}(x^4) = 4x^{3}$
- For the second term $6x^{- rac{1}{2}}$ :
  
  $\frac{d}{dx}(6x^{- rac{1}{2}}) = 6 \cdot -\frac{1}{2} x^{- rac{3}{2}} = -3x^{- rac{3}{2}}$
Combine the results:

$\frac{dy}{dx} = 4x^{3} - 3x^{- rac{3}{2}}$

Therefore, the simplified form is:

$\frac{dy}{dx} = 4x^{3} - \frac{3}{x^{\frac{3}{2}}}$

Step 2

(b) $\int y \, dx$

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Answer

To calculate the integral $\, \int y \, dx$ where $y = x^4 + 6x^{- rac{1}{2}}$ , we integrate each term separately:

The integral of $x^4$ is:

$\int x^4 \, dx = \frac{x^5}{5}$
The integral of $6x^{- rac{1}{2}}$ is:

$\int 6x^{- rac{1}{2}} \, dx = 6 \cdot \int x^{- rac{1}{2}} \, dx = 6 \cdot 2x^{\frac{1}{2}} = 12x^{\frac{1}{2}}$
Combine the results and add the constant of integration $C$ :

$\int y \, dx = \frac{x^5}{5} + 12x^{\frac{1}{2}} + C$

Hence, the final answer is:

$\int y \, dx = \frac{x^5}{5} + 12x^{\frac{1}{2}} + C$

Given that $y = x^4 + 6x^{- rac{1}{2}}$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \: dx$ - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 1

Worked Solution & Example Answer:Given that $y = x^4 + 6x^{- rac{1}{2}}$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \: dx$ - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 1

(a) $\frac{dy}{dx}$

(b) $\int y \, dx$

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Given that $y = x^4 + 6x^{-rac{1}{2}}$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \: dx$ - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 1

Worked Solution & Example Answer:Given that $y = x^4 + 6x^{-rac{1}{2}}$, find in their simplest form (a) $\frac{dy}{dx}$ (b) $\int y \: dx$ - Edexcel - A-Level Maths Pure - Question 2 - 2012 - Paper 1