High School

For the following exercises, find the differential of the function:

68. \( y = 3x^4 + x^2 − 2x + 1 \)

69. \( y = x \cos x \)

70. \( y = \sqrt{1 + x} \)

71. \( y = \frac{x^2 + 2}{x} − 1 \)

Answer :

Final answer:

To find the differentials of the given functions, we can use the rules of differentiation. The differentials for the functions are:

[tex]1. dy = 12x^3 + 2x - 2, 2. dy[/tex]

[tex]1/2)(1 + x)^(-1/2), and 4. dy = (2x - x^2 - 2)/(x - 1)^2.[/tex]

Explanation:

To find the differentials of the given functions, we can use the rules of differentiation. Let's find the differentials for each function:

  1. y = 3x4 + x2 - 2x + 1 (Differential: dy = 12x3 + 2x - 2)
  2. y = xcos(x) (Differential: dy = cos(x) - xsin(x))
  3. y = sqrt(1 + x) (Differential: dy = (1/2)(1 + x)-1/2)
  4. y = (x2 + 2)/(x - 1) (Differential: dy = (2x - x2 - 2)/(x - 1)2)

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