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Analisi 1: Derivate Parte 1
f(x) = 5x⁴ - 2x⁶ + 3
f'(x) = 20x³ - 12x⁵
f(x) = x²/(x+2)
f'(x) = (2x(x+2) - x²)/(x+2)² = 2x²+4x-x²/(x+2)² = x²+2x/(x+2)²
f(x) = x²/√(x²+2)
f'(x) = x((√(x²+2))'-((√(x²+2))')x)/(√(x²+2))² = 2x√(x²+2) - x³/√(x²+2)
= (4x(x²+2) - x³)/2√(x²+2) = x³+4x/(x²+2)^(3/2)
f(x) = extan³(x) + e3xtan²(x) cos³(x) = etan²(x) tan(x) + cos³(x)
f(x) = x ln(x) + x⁴
f'(x) = ln(x) + x(1/x) + 4x³ = ln(x) + 4x³ + 1
f(x) = exsin(x)
f'(x) = exsin(x) + excos(x)
f(x) = e-x(sin(x) + cos(x))
f'(x) = -e-x(sin(x) + cos(x)) + ex(cos(x) + sin(x)) = ex(sin(x) + cos(x)) + cos(x) + sin(x)
f(x) = cos(x) + 1/x + sin(x)
f'(x) = -sin(x)(x + sin(x)) - (cos(x) + 1)(1 + cos(x))/(x + sin(x))²
= (-sin(x)(x) - cos(x)(1) - 2)(x + sin(x) - 2cos(x) - 1)/(x + sin(x))²
9) f(x) = sin(x2)/x+1
f'(x) = (x2 + x)(cos(x2)(x+1) - sin(x2)) /(x+1)2
10) f(x) = ln(2 + ex)
f'(x) = 1/ √x + ex
= ex/√2√x + ex
11) f(x) = sin 1/x2 + x + 1
f'(1)(x) = cos(1/x2 + x + 1) (-1/(x2 + x + 1)2) (2x + 1)
= cos(1/x2 + x + 1) (-2x + 1/(x2 + x + 1)2)
12) f(x) = ln(x + 3/x + 5)
f'(x) = (x + 3/1)(2) / (x + 5)2 - 2ln(x + 3)(2x) / (x + 5)2 - 2xln(x + 3)
13) f(x) = x sin(x - 5 - 7)
f'(x) = sin(x - 5 - 7) + x cos(x - 5 - 7)(x + 7) = sin(x - 5 - 7) + 5x5 cos(x - 5 - 7)
14) f(x) = cos β (sinβ(sinβ(x)))/cos β(x)
f'(x) = sin β (sinβ(x))cosβ(x)cosβ(x)/(cosβ(x))2 = sin β (sinβ(x)cosβ2(x) - cosβ(sinβ(x))sinβ(x)x
15) f(x) = 2ctam [sin(2πm(x))]
f'(x) = 1/1 + 5ct(sin(x)) cos (2πm(x))(x) ( -1/x) = 2x cos(2πm(x)) / x + x sin(2πm(x))