Определение производных

1) f (x) = 3x²-1/x+1

2) f (x) = 2x+1/x²

3) f (x) = x²/2x-1

4) f (x) = x²-4/2x³-1

1) f (x) = (3x²-1) / (x+1)

f' (x) = [6x (x+1) - 1 * (3x ²-1) ] / (x+1) ² = (6x²+6x-3x²+1) / (x+1) ² = (3x²+6x+1) / (x+1) ²

если не дробь, то

f' (x) = 6x+1/x ²

2) f (x) = (2x+1) / x²

f' (x) = (2x ²-4x²-2x) / x^4 = (-2x²-2x) / x^4 = (-2x-2) / x³

если не дробь, то

f' (x) = 2-2/x ³

3) f (x) = x² / (2x-1)

f' (x) = (4x ²-2x-2x²) / (2x-1) = (2x²-2x) / (2x-1) ²

4) f (x) = (x²-4) / (2x³-1)

f' (x) = (4x^4-2x-6x^4+24x ²) / (2x³-1) ²=24x²-2x-2x^4) / (2x³-1) ²

1) f' (x) = 6x (x+1) - (3x²-1) * 1 / (x+1) ² = 6x²+6x-3x²+1 / (x+1) ²=

=3x²+6x+1 / (x+1) ²

2) f' (x) = 2*x² - (2x+1) * 2x/x⁴=2x²-4x²-2x/x⁴=-2x²-2x/x⁴=-2x-2/x³

3) f' (x) = 2x * (2x-1) - x²*2 / (2x-1) ²=4x²-2x-2x² / (2x-1) ²=2x²-2x / (2x-1) ²

4) f' (x) = 2x * (2x ³-1) - (x²-4) * 6x² / (2x³-1) ²=4x⁴-2x-6x⁴+24x² / (2x³-1) ²=

=24x ²-2x-2x⁴ / (2x³-1) ²