Analisi matematica 1 - Derivate

Esercizi derivate

g(x) = 3x + 4 ln x - 2e^x + 3 cos x
g'(x) = 3 + ⁴/_x - 2e^x - 3 sin x

g(x) = ⁵√₃x + ⁷√₃x - ¹/_x = x^3/5 + x^2/3 - x^-1
g'(x) = ⁷/₃x^-2/₃ + 2 x^-1/₃ - ¹/₅x^-6/₅ = ³/₅√_x² + ²/₃√₃x + ¹/₅√₅x

g(x) = (x³ + 2x² + x) . ln x
g'(x) = (3x² + 4x + 1) ln x + ^{x3 + 2x2 + x}/_x = (3x² + 4x + 1) ln x + x² + 2x + 1

g(x) = (2x + 3) (x² + 3x - 1)
g'(x) = 2 (x² + 3x - 1) + (2x + 3)(2x + 3) = 2x³ + 6x - 2 + 4x² + 9 + 12x = 6x² + 18x + 7

g(x) = ^{x2 - 3x + 5}/_{x² - 1}
g'(x) = ^{(2x - 3)(x2 + 1) - (x2 - 3x + 5)(2x)}/_{(x² - 1)²} = ^{4x2 - 2x - 3x2 + 3 - 2x2 + 6x - 10x}/_{(x² - 1)²} = ^{3x2 - 12x + 3}/_{(x² - 1)²} = ^{3(x2 - 4x + 11)}/_{(x² - 1)²}

Esercizi derivate

g(x) = 3x + 4 ln x - 2e^x + 3 cos x
g'(x) = 3 + ⁴/_x - 2e^x - 3 sin x

g(x) = ⁵√_x + ³√_x - ¹/_x = x^3/5 + x^2/3 - x^-1
g'(x) = ³/₅ x^-2/5 + ²/₃ x^-1/3 - ¹/_x² = ³/₅ x^-2/5 + ²/₃ x^-1/3 + ¹/_x⁵√x

g(x) = (x³ + 2x² + x) . ln x
g'(x) = (3x² + 4x + 1) ln x + ^{x3 + 2x2 + x}/_x = (3x² + 4x + 1) ln x + x² + 2x + 1

g(x) = (2x + 3) (x² + 3x - 1)
g'(x) = 2 (x² + 3x - 1) + (2x + 3) (2x + 3)
= 2x² + 6x - 2 + 4x² + 9 + 12x = 6x² + 18x + 7

g(x) = ^{x2 - 3x + 5}/_{x² - 1}
g'(x) = ^{(2x - 3) (x2 - 1) - (x2 - 3x + 5) (2x)}/_{(x² - 1)²} = ^{4x2 - 2x - 3x2 + 5}/_{(x² - 1)²} = ^{3x2 - 12x + 3}/_{(x² - 1)²} = ^{3(x2 - 4x + 11)}/_{(x² - 1)²}

Derivate

17)

g(x) = ^{x3 - 2lnx}/_x
g'(x) = (3x² - ²/_x)x² - (x³ - 2lnx) ⋅ 2x = ^{3x4 - 2x - 2x4 + 4xlnx}/_x² = ^{x4 - 2x + 4xlnx}/_x² = x(x³ - 2 + ^4lnx/_x²)

18)

g(x) = ^{3x2 - 2x + 1}/_{x² - 2}
g'(x) = ^{(6x - 2)(x2 - 2) - (3x2 - 2x + 1)(2x)}/_{(x² - 2)²} = ^{6x3 - 12x - 2x3 + 4 - 6x3 + 4x2 - 2x}/_{(x² - 2)²} = ^{2x2 - 14x + 4}/_{(x² - 2)²}

20)

g(x) = (x² - 3x - 5) ⋅ (3x² - 2x + 1) + ^{x2 + 1}/_{3(x² - 1)}
g'(x) = (2x - 3)(3x² - 2x + 1) + (x² - 3x - 5)(6x - 2) + ^{6x(x2 - 1) - (x2 + 1)(6x)}/_{9(x² - 1)²} = 6x³ - 4x² + 2x - 9x² + 6x - 3 + 6x³ + ^4x2 - 18x + 10 + ^{6x3 + 6x}/_{9(x² - 1)²} = 12x³ - 33x² - 16x + 7 - ^4x/_{3(x² - 1)²}

27)

g(x) = (3x² - 2x + 1)⁵
g'(x) = 5(3x² - 2x + 1)⁴(6x - 2)

29)

g(x) = x - √(1 - x²) = x - (1 - x²)^1/2
g'(x) = 1 - 1/2 (1 - x²)^-1/2 (-2x) = 1 + x / √(1 - x²)

33)

g(x) = √((3x + 7) / (8 - x)) = ((3x + 7) / (8 - x))^1/2
g'(x) = 1/2 ((3x + 7) / (8 - x))^-1/2 ((3(8 - x) - (3x + 7)(-1)) / (8 - x)²) = 1/2 √((8 - x) / (3x + 7)) ((24 - 3x + 3x + 7) / (8 - x)²) = 1/2 √((8 - x) / (3x + 7)) ((31) / (8 - x)²)

34)

g(x) = √((2x - 9) / (1 - x)) = ((2x - 9) / (1 - x))^1/2 = 1/2 √((1 - x) / (2x - 9)) ((2 - 2x + 2x - 9) / (1 - x)²)

38)

g(x) = e^_x2+5/_x+1
g'(x) = e^_x2+5/_x+1 ⋅ (_{2x²+2x-x²-5}/_(x+1)²) = e^_x2+5/_x+1 ⋅ (_x²+2x-5/_(x+1)²)

40)

g(x) = x³ ⋅ e^-x3
g'(x) = 3x² e^-x3 + x³ e^-x3 ⋅ (-3x²) = 3x² e^-x3 (1 - x³)

42)

g(x) = ln (x²+1)
g'(x) = ¹/_x²+1 ⋅ 2x = ^2x/_x²+1

43)

g(x) = ln (^5x+4/_x-3)
g'(x) = ^5x-15-5x-4/_(x-3)x = ^-19/_(x-3)(5x+4)

44)

g(x) = xln³√x + sin 2x
g'(x) = cos 3√x ⋅ ^x/₃x^-3/₂ - sin 2x ⋅ 2 = ^{cos 3√x}/_3√x² - 2 sin 2x

a)

g(x) = sin ₍ ^{3x - 2}/_{2x + 7 )}
g'(x) = cos ₍ ^{3x - 2}/_{2x + 7 )} . (^{6x + 21 - 6x + 4}/_{(2x + 7)²})

1)

g(x) = ln (ln x)
g'(x) = ¹/_{ln x} . ¹/_x

2)

g(x) = e^√(x2+3x+1)
g'(x) = e^√(x2+3x+1) . ¹/_{2 √(x²+3x+1)} . (2x + 3)

4)

g(x) = ln (√6x)
g'(x) = ¹/_√6x . ¹/_{2 √6x} . (-sin x) = -^{sin x}/_{2 6x}

Più difficili ↓

23)

y = ^{3\ln 2x}/_{2\cos x (1 + \sin²x)} = ^{6\ln x \cos x}/_{2\cos x (1 + \sin²x)} = ^{3\ln x}/_{1 + \sin²x}
y' = ^{3\cos x (1 + \sin2x) - (3\ln x)(2\sin x \cos x)}/_{(1 + \sin²x)²}
y' = ^{3\cos x + 3\cos x \sin2x - 6\ln x \sin2x \cos x}/_{(1 + \sin²x)²}
y' = ^{3\cos x - 3\sin2x \cos x}/_{(1 + \sin²x)²} = ^{3\cos x (1 - \sin2x)}/_{(1 + \sin²x)²} = ^3\cos3x/_{(1 + \sin²x)²}

24)

y = √(1 - x²) - x \arccos x
y' = ¹/_{√(1 - x²)} • (-2x) - \arccos x - x (-¹/_{√(1 - x²)})
y' = -2x/_{√(1 - x²)} - \arccos x + x/_{√(1 - x²)} = -\arccos x

25)

y = \ln (^{√(x2 + 1) + x}/_{√(x² + 1) - x}) = \ln (^{√(x2 + 1) + x}/_{√(x² + 1) - x} • ^{√(x2 + 1) + x}/_{√(x² + 1) + x}) = \ln (^{(√(x2 + 1) + x)2}/_{x² + 1}) = 2 \ln (√(x² + 1) + x)