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y = 5x^4 - 6x^3 + 2x - 3 (a) Find \( \frac{dy}{dx} \) giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 2
Question 6
y = 5x^4 - 6x^3 + 2x - 3 (a) Find \( \frac{dy}{dx} \) giving each term in its simplest form. (b) Find \( \frac{d^2y}{dx^2} \)
Worked Solution & Example Answer:y = 5x^4 - 6x^3 + 2x - 3 (a) Find \( \frac{dy}{dx} \) giving each term in its simplest form - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 2
Step 1
Find \( \frac{dy}{dx} \) giving each term in its simplest form.
Answer
To find ( \frac{dy}{dx} ), we differentiate each term of the function ( y = 5x^4 - 6x^3 + 2x - 3 ) using the power rule:
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For ( 5x^4 ): [ \frac{d}{dx}(5x^4) = 20x^3 ]
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For ( -6x^3 ): [ \frac{d}{dx}(-6x^3) = -18x^2 ]
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For ( 2x ): [ \frac{d}{dx}(2x) = 2 ]
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For the constant ( -3 ): [ \frac{d}{dx}(-3) = 0 ]
Combining these results gives: [ \frac{dy}{dx} = 20x^3 - 18x^2 + 2 ]
Step 2
Find \( \frac{d^2y}{dx^2} \)
Answer
To find ( \frac{d^2y}{dx^2} ), we differentiate ( \frac{dy}{dx} = 20x^3 - 18x^2 + 2 ) again:
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For ( 20x^3 ): [ \frac{d}{dx}(20x^3) = 60x^2 ]
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For ( -18x^2 ): [ \frac{d}{dx}(-18x^2) = -36x ]
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For ( 2 ): [ \frac{d}{dx}(2) = 0 ]
Thus, we have: [ \frac{d^2y}{dx^2} = 60x^2 - 36x ]