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Given that $y = 6x - \frac{4}{x^2}$, $x \neq 0$ - Edexcel - A-Level Maths Pure - Question 4 - 2005 - Paper 1

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Question 4

Given-that-$y-=-6x---\frac{4}{x^2}$,-$x-\neq-0$-Edexcel-A-Level Maths Pure-Question 4-2005-Paper 1.png

Given that $y = 6x - \frac{4}{x^2}$, $x \neq 0$. (a) find $\frac{dy}{dx}$. (b) find $\int y \; dx$.

Worked Solution & Example Answer:Given that $y = 6x - \frac{4}{x^2}$, $x \neq 0$ - Edexcel - A-Level Maths Pure - Question 4 - 2005 - Paper 1

Step 1

find $\frac{dy}{dx}$

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Answer

To find the derivative of yy, we apply the rules of differentiation.

Given: y=6x−4x2y = 6x - \frac{4}{x^2}

We differentiate term by term:

  1. The derivative of 6x6x is 66.

  2. For the term −4x2-\frac{4}{x^2}, we can rewrite it as −4x−2-4x^{-2}.

    Therefore, the derivative is: dydx=6+8x−3\frac{dy}{dx} = 6 + 8x^{-3}

    Combining it, we have: dydx=6+8x3\frac{dy}{dx} = 6 + \frac{8}{x^3}

Step 2

find $\int y \; dx$

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Answer

To find the integral of yy, we integrate the function term by term.

Starting with: y=6x−4x2y = 6x - \frac{4}{x^2}

We can express this in a more integrable form: y=6x−4x−2y = 6x - 4x^{-2}

Now, we integrate:

  1. The integral of 6x6x is: ∫6x dx=3x2+c\int 6x \, dx = 3x^2 + c

  2. The integral of −4x−2-4x^{-2} is: ∫−4x−2 dx=4x−1=4x\int -4x^{-2} \, dx = 4x^{-1} = \frac{4}{x}

Combining both results: ∫y dx=3x2+4x−1+c\int y \, dx = 3x^2 + 4x^{-1} + c

Thus, we have: ∫y dx=3x2+4x+c\int y \, dx = 3x^2 + \frac{4}{x} + c

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