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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 3 - 2012 - Paper 1

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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 3-2012-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 3 - 2012 - Paper 1

Step 1

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

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Answer

To find the derivative of the function, we apply the power rule of differentiation:

  1. Differentiate each term:

    • For x4x^4: The derivative is 4x34x^{3}.
    • For 6x^{- rac{1}{2}}: The derivative is 6⋅−12x−32=−3x−326 \cdot -\frac{1}{2} x^{-\frac{3}{2}} = -3x^{-\frac{3}{2}}.

  2. Combine the results:

    dydx=4x3−3x−32\frac{dy}{dx} = 4x^{3} - 3x^{-\frac{3}{2}}

Thus, the final answer is:

dydx=4x3−3x−32\frac{dy}{dx} = 4x^{3} - 3x^{-\frac{3}{2}}

Step 2

(b) $\int y \, dx$

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Answer

To find the integral of yy with respect to xx, we integrate each term separately:

  1. Integrate x4x^4:

    ∫x4 dx=x55\int x^4 \,dx = \frac{x^{5}}{5}

  2. Integrate 6x^{- rac{1}{2}}:

    \int 6x^{- rac{1}{2}} \,dx = 6 \cdot 2x^{\frac{1}{2}} = 12x^{\frac{1}{2}}

  3. Combine the integral results along with a constant of integration CC:

    ∫y dx=x55+12x12+C\int y \, dx = \frac{x^{5}}{5} + 12x^{\frac{1}{2}} + C

Thus, the final answer is:

∫y dx=x55+12x12+C\int y \, dx = \frac{x^{5}}{5} + 12x^{\frac{1}{2}} + C

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