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Given that $y = 6x - \frac{4}{x^{2}}$, $x \neq 0$, (a) find \( \frac{dy}{dx} \) - Edexcel - A-Level Maths Pure - Question 4 - 2005 - Paper 1
Question 4
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 $y = 6x - \frac{4}{x^{2}}$, $x \neq 0$, (a) find \( \frac{dy}{dx} \) - Edexcel - A-Level Maths Pure - Question 4 - 2005 - Paper 1
Step 1
find \( \frac{dy}{dx} \)
Answer
To find ( \frac{dy}{dx} ), we use the formula for the derivative of the function given by:
[ y = 6x - 4x^{-2} ]
The derivative is computed as follows:
[ \frac{dy}{dx} = \frac{d(6x)}{dx} - \frac{d(4x^{-2})}{dx} ]
The derivative of ( 6x ) is ( 6 ), and using the power rule, the derivative of ( -4x^{-2} ) becomes:
[ -4 \cdot (-2) x^{-3} = \frac{8}{x^{3}} ]
Therefore,
[ \frac{dy}{dx} = 6 + \frac{8}{x^{3}} ]
Step 2
find \( \int y \, dx \)
Answer
To find the integral of ( y ), we compute:
[ \int y , dx = \int (6x - 4x^{-2}) , dx ]
Integrating term by term gives us:
- ( \int 6x , dx = 3x^{2} + C_1 )
- ( \int -4x^{-2} , dx = 4x^{-1} + C_2 = \frac{4}{x} + C_2 )
Therefore, combining these results, we have:
[ \int y , dx = 3x^{2} + 4\frac{1}{x} + C ]
Where ( C ) is the constant of integration.