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

Question

f(x) = (x^2)/(1 + 8x)

Answer 1

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