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Given that $y = 2x^{6} + 7 + \frac{1}{x^{3}}$, $x \neq 0$, find, in their simplest form, (a) \( \frac{dy}{dx} \) - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 1
Question 5
Given that $y = 2x^{6} + 7 + \frac{1}{x^{3}}$, $x \neq 0$, find, in their simplest form, (a) \( \frac{dy}{dx} \). (b) \int y \, dx.
Worked Solution & Example Answer:Given that $y = 2x^{6} + 7 + \frac{1}{x^{3}}$, $x \neq 0$, find, in their simplest form, (a) \( \frac{dy}{dx} \) - Edexcel - A-Level Maths Pure - Question 5 - 2011 - Paper 1
Step 1
(a) \( \frac{dy}{dx} \)
Answer
To find ( \frac{dy}{dx} ), we will differentiate each term of the function separately:
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Differentiate ( 2x^{6} ): [ \frac{d}{dx}(2x^{6}) = 12x^{5} ]
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Differentiate ( 7 ): [ \frac{d}{dx}(7) = 0 ]
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Differentiate ( \frac{1}{x^{3}} ): Using the power rule, this term can be rewritten as ( x^{-3} ): [ \frac{d}{dx}(x^{-3}) = -3x^{-4} = -\frac{3}{x^{4}} ]
Combining all the results, we have: [ \frac{dy}{dx} = 12x^{5} + 0 - \frac{3}{x^{4}} = 12x^{5} - \frac{3}{x^{4}} ]
Step 2
(b) \int y \, dx
Answer
To integrate ( y = 2x^{6} + 7 + \frac{1}{x^{3}} ), we will integrate each term separately:
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Integrate ( 2x^{6} ): [ \int 2x^{6} , dx = \frac{2}{7}x^{7} = \frac{2}{7}x^{7} ]
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Integrate ( 7 ): [ \int 7 , dx = 7x ]
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Integrate ( \frac{1}{x^{3}} ): This can be rewritten as ( x^{-3} ): [ \int x^{-3} , dx = \frac{x^{-2}}{-2} = -\frac{1}{2x^{2}} ]
Combining these results gives: [ \int y , dx = \frac{2}{7}x^{7} + 7x - \frac{1}{2x^{2}} + C ]