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Find f '(x) and f ''(x).?

+3 votes
f(x) = (x^2)/(1 + 8x)
asked Feb 19, 2013 in CALCULUS by futai Scholar

1 Answer

+2 votes

f(x) = x2/(1 + 8x)

Apply derivative with respect to x' each side.

f'(x) = (d/dx)[x2/(1 + 8x)]

The Quotient Rule: (d/dx)(u/v) = [(vu' - uv')]/v2

Where u = x 2⇒ u' = 2x and v = (1+8x) ⇒ v' = 0+8 = 8

f'(x) = [(1+8x)(2x) - (x2)(8)]/(1 + 8x)2

f'(x) = [2x + 16x2- 8x2]/(1 + 8x)2

f'(x) = [2x + 8x2]/(1 + 8x)2

Again apply derivative with respect to x' each side.

The Quotient Rule: (d/dx)(u/v) = [(vu' - uv')]/v2

Where u = 2x + 8x 2⇒ u' = 2 + 16x and v = (1 + 8x)2 ⇒ v' = 2(1 + 8x)(8) = 16(1 + 8x)

f''(x) = [(1 + 8x)2(2 + 16x) - (2x + 8x2)16(1 + 8x)]/(1 + 8x)4

f''(x) = [(1 + 8x)22(1 + 8x) - 2x(1 + 4x)16(1 + 8x)]/(1 + 8x)4

f''(x) = [2(1 + 8x)3- 32x(1 + 4x)(1 + 8x)]/(1 + 8x)4

answered Feb 19, 2013 by richardson Scholar

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