Divide. −4 3/5 ÷ 1 1/5 Enter your answer as a mixed number, in simplified form, in the box.
1 answer:
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The second derivative at the point (2,2) is 34/9
<u>Explanation:</u>
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2x⁴ = 4y³
2x⁴ - 4y³ = 0
We first need to find dy/dx and then d²y/dx²
On differentiating the equation in terms of x
dy/dx = d(2x⁴ - 4y³) / dx
We get,
dy/dx = 2x³/3y²
On differentiating dy/dx we get,
d²y/dx² = 2x²/y² + 8x⁶/9y⁵
d²y/dx² = 34/9
Therefore, the second derivative at the point (2,2) is 34/9
Start with the answer format we want, and work your way toward forming a single fraction like so
Compare that last expression to (2x+1)/(x+2). Notice how the ax and 2x match up, so a = 2 must be the case.
Then we have 2a+b as the remaining portion in the numerator. Plugging in a = 2 leads to 2a+b = 2*2+b = 4+b. Set this equal to the +1 found in (2x+1)/(x+2) to have the terms match.
So, 4+b = 1 leads to b = -3
Therefore, a = 2 and b = -3
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An alternative route:
I added and subtracted 4 in the third step so that I could form 2x+4, which then factors to 2(x+2). That way I could cancel out a pair of (x+2) terms toward the very end.
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Other alternative methods involve synthetic division or polynomial long division. They are slightly separate but related concepts.
