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Find the second derivative of the given function.


f(x) = \frac{x {}^(2) + 2x - 1}{ x}


User Advanced Customer
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2 Answers

21 votes
21 votes

Answer:

= -2/x^3

Explanation:

Possible derivation:

d/dx(d/dx((x^2 + 2 x - 1)/x))

Distribute 1/x through x^2 + 2 x - 1:

= d/dx((d/dx(2 - 1/x + x)))

Differentiate the sum term by term and factor out constants:

= d/dx((d/dx(2) - d/dx(1/x) + d/dx(x)))

The derivative of 2 is zero:

= d/dx(0 - d/dx(1/x) + d/dx(x))

Use the power rule, d/dx(x^n) = n x^(n - 1), where n = -1.

d/dx(1/x) = d/dx(x^(-1)) = -x^(-2):

= d/dx(0 - -1/x^2 + d/dx(x))

The derivative of x is 1:

= d/dx(0 - -1/x^2 + 1)

Differentiate the sum term by term:

= d/dx(1) + d/dx(1/x^2)

The derivative of 1 is zero:

= d/dx(1/x^2) + 0

Simplify the expression:

= d/dx(1/x^2)

Use the power rule, d/dx(x^n) = n x^(n - 1), where n = -2.

d/dx(1/x^2) = d/dx(x^(-2)) = -2 x^(-3):

Answer: = -2/x^3

User Marverix
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20 votes
20 votes
The second derivative would be -2/x^3 (the whole fraction is negative)
User VAIRIX
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