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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 7 - 2012 - Paper 2
Question 7
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 7 - 2012 - Paper 2
Step 1
Find \(\frac{dy}{dx}\)
Answer
To find (\frac{dy}{dx}), we will differentiate each term of the function with respect to (x).
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Differentiate (5x^4): [ \frac{d}{dx}(5x^4) = 20x^3 ]
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Differentiate (-6x^3): [ \frac{d}{dx}(-6x^3) = -18x^2 ]
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Differentiate (2x): [ \frac{d}{dx}(2x) = 2 ]
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The derivative of (-3) is 0 since it's a constant.
Combining all these, we get:
[ \frac{dy}{dx} = 20x^3 - 18x^2 + 2 ]
Step 2
Find \(\frac{d^2y}{dx^2}\)
Answer
Now we will differentiate (\frac{dy}{dx} = 20x^3 - 18x^2 + 2) to find (\frac{d^2y}{dx^2}).
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Differentiate (20x^3): [ \frac{d}{dx}(20x^3) = 60x^2 ]
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Differentiate (-18x^2): [ \frac{d}{dx}(-18x^2) = -36x ]
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The derivative of (2) is 0.
Therefore, we have:
[ \frac{d^2y}{dx^2} = 60x^2 - 36x ]