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Expert: - 11/3/2009


Question
my problems (set 1):
In Exercises 4, 6, 10, 15, 17, 18 use the result of these problems to find dy/dx or its derivative.
4. y=(1/2)((x^4)+7).
6. y=(radical[2])(x)+(1/(radical[2])).
10. y=(1/a)(x²+(1/b)(x)+c) (a, b, c constant).
15. y=(3x²+6)(2x-(1/4)).
17. y=(x³+7x²-8)((2x^-3)+(x^-4)).
18. y=((1/x)+(1/x²))(3x³+27).
For Exercise 43, find dy/dx or its derivative using the extension of the product rule.
43. Find d/dx [(2x+1)(1+(1/x))((x^-3)+7)].

Answer
4. y = x^4/2 + 7/2, so y' = 5x^3/2 = 2x^3.

6. y = √2•x + 1/√2, so y' = √2.

10. y = (1/a)(x² + (1/b)x + c), so y' = 2x/a + 1/(ab).

15. y = (3x²+6)(2x - 1/4).  This is the same as y ' = 6x³ - 0.75x² + 6x - 15.
Therefore y' = 18x² - 1.5x + 6.

17) y = 2 + 14/x - 10/x^3 + 1/x + 7/x^2 - 5/x^3.
That is, y = 2 + 15/x + 7/x^2 - 10/x^3 - 5/x^4.
Therefore y' = -15/x^2 - 14/x^3 - 30/x^4 -20/x^5.

43. If y = (2x+1)(1 + 1/x)(1/x^3 + 7), that is the same as
y = (2x+1)(x+1)(7x^3+1)/x^3.
y = (2x^2+3x+1)(7x^3+1)/x^3.
y = (14x^5+21x^4+7x^3 + 2x^2+3x+1)/x^3.

From here, the quotient rule will tell us y.
y' = ((x^3)(70x^4+85x^3+21x^2+4x+3)-(14x^5+21x^4+7x^3 + 2x^2+3x+1)3x^2)/x^6.
This simplifies to y' = (70x^7 + 85x^6 + 7x^5 + 4x^4 - 4x^3 + 3x^2 + 3x + 1)/x^6.

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