Derivate delle funzioni goniometriche inverse, esercizi svolti

Analisi matematica

Le derivate

Esercizi svolti

Derivata delle funzioni inverse delle funzioni goniometriche

• D[arcsin x] = ¹⁄_√1-x²
• D[arccos x] = - ¹⁄_√1-x²
• D[arctan x] = ¹⁄_1+x²
• D[arccot x] = - ¹⁄_1+x²

Calcola la derivata delle seguenti funzioni

• Y = 2 arcsin x + arccos x
• Y = (1 + x²) arctan x
• Y = x arccos √(1-x²)
• Y = x - √1-x² arscin x
• Y = arcsin x²
• Y = arccos 4x
• Y = arctan x³
• Y = ¹⁄₃ arccos x²
• Y = arctan e^x
• Y = (arctan x)⁴
• Y = 2 arccos ^x⁄₂
• Y = 2 arcsin √x
• Y = ln (arcsin x)
• Y = 4x - arctan x
• Y = arctan x + ¹⁄_{(x - 1)2}
• Y = arcsin x + √1-x²
• Y = arcsin x - arccos x
• Y = arccos x arccos x
• Y = arcsin √x arccos x
• Y = arccos x arsin x
• Y = arcsin x x arccos x
• Y = arccos x ^x-1⁄_x+1
• Y = arccos (1 - x²)
• Y = arcsin 2y 0 \), è la derivata di una sola delle seguenti funzioni. Quale?
• A \( y = x + \frac{1}{x} \)
• B \( y = x - \frac{1}{x} \)
• C \( y = x + \ln x \)
• D \( y = x \cdot \ln x \)

491 La funzione \( y = \frac{1 - \ln x}{x^2} \), con \( x > 0 \), è la derivata di tutte queste funzioni, tranne una. Quale?

• A \( y = \frac{\ln x + 2x}{x} \)
• B \( y = 1 + \frac{\ln x}{x} \)
• C \( y = \frac{\ln x}{x} \)
• D \( y = x + \frac{\ln x}{x} \)

M^oh61

Y = 2·arcsen x + arccos x

Y'_d = 2 · 1/√(1-x²) - 1/√(1-x²)

Y' = 2·1/√(1-x²) - 1/√(1-x²) = 1/√(1-x²) ;

M^oh62

Y_q = ln x - arctan x

Y'_d = 4 · 1/1+x²

Y' = 4(1+x²) · 1/1+x²

Y' = 4 + 4x² - 4/1 + x²

Y' = 4x² + 3/1 + x² ;

M^oh63

Y_q = (1+x²)·arctan(x)

Y' = 2x·arctan(x) + (1+x²)² · 1/1+x²

Y' = 2x·arctan(x+1) ;

Y = x · arc cos x - √1-x²

Y' = 1 · arc cos x + x · ⎡ - 1 / √1-x²⎤ = - 1 / 2√1-x² (-2x)

Y' = arc cos x - x / √1-x² = + 2x / 2√1-x²

Y' = arc cos x ;

Y = arc tan x + 1/2 arc cot x

Y' = 1 / 1+x² + 1/2 ( - 1 / 1+x² )

Y' = 2 - 1 / 2(1+x²)

Y' = 1/x 1/1+x² = x/1+x² ;

Y = x - √1-x² · arc tan x

Y' = 1 - ⎡ -1 / 2√1-x²⎤ arc sin x + √1-x² ⎡ 1 / √1-x²⎤

Y' = 1 ⎡ x · arc sin x + 1 / √1-x²⎤ -

Y' = 1 / + x · arc sin x / √1-x² = -

Y' = x / √1-x² ;

M₄₆₇

Y = arc sin x + ¹/_{1 + x²}

Y' = ¹/_{√1 - x²} + ^{0 (1 + x2)1 - 1(2x)}/_{(1 + x²)²}

Y' = ¹/_{√1 - x²} - ^2x/_{(1 + x²)²}

Y' = ¹/_{√1 - x²} - ^2x/_{(1 + x²)²}

Y' = ^x2/_{(1 + x²)²} + 1

Y' = ^{(x - 1)2}/_{(1 + x²)²} ;

M₄₆₈

Y = arc sin x + √1 - x²

Y' = ¹/_{√1 - x²} + ^{√1 - x2}

Y' = ¹/_{√1 - x²} - ^x/_{√1 - x²}

Y' = ^{1 - x}/_{√1 - x²} ;

M 969

Y = arcsenx - arccosx

Y' = ^-1/_√(1-x²) - arcsenx + arcsenx ^-1/_√(1-x²)

Y = ¹/_√(1-x²) - arccosx - arcsenx

Y' = ^{-arccosx - arcsenx}/_√(1-x²)

M 970

Y = ^arccosx/_arcsenx

Y' = ¹/_√(1-x²) arcsenx - arccosx 1/_{√(1-x²)             (arcsenx)²}

Y' = ^{-arcsenx - arccosx}/_√(1-x²) / _(arcsenx)²

Y' = ^{-arcsenx + arccosx}/_√(1-x²) ¹/_(arcsenx)²

Y = arcsin(x²)

Y' = ¹/_√1-(x²)² · 2x

Y' = ^2x/_√1-x⁴

Y = arccos 4x

Y' = -¹/_√1-(4x)² · 4

Y' = -⁴/_√1-16x²

Y = arctan x³

Y' = -¹/_1+(x³)² · 3x²

Y' = ¹/_1+x⁶ · 3x²

Y' = ^3x2/_1+x⁶

Y = ²/₃ arccos² x

N. 3 arccosⁿ x ≠ arccos xⁿ

Arccosⁿ x = (arccos x)² = (β(x))²

D [(ω)ⁿ] = n (ω)^n-1 · D ω = n [(β(x))^n-1 * β'_(x)]

Y' = -^1/3 2 arccos x · -¹/_{√1 - x²}

Y' = -^2/3 arccos x / _{√1 - x²}

Y = arctan e^x

Y' = -^1/(1+(e^x)²) · e^x

Y' = ^1/(1+ e^x2) · e^x

Y' = e^x / (e^2x + 1)

Y = (arctan x)⁴

Y' = 4 [arctan x]³ · 1/ _{1 + x²}

Y' = 4 (x arctan x)³ = 4 · arctan³(x) / _{1 + x²}

Y = ²/₃ arccos² x

N^o 475

N. 3 arccos x ≠ arccos x²

Arccos x = (arccos x)² = (β(x))²

D ([α(x)]ⁿ) = n [α(x)]^n-1 ⋅ D [α(x)]

Y' = -¹/₃ 2 arccos x - -¹/_√1-x²

Y' = -²/₃ arccos x / √1-x²

N^o 476

Y = arctan e^x

Y' = -¹/_1+(e^x)² ⋅ e^x

Y' = ¹/_1+e^2x ⋅ e^x

Y' = ^ex/_{e^2x + 1}

N^o 477 → [β(x)]ⁿ

Y = (areing x)⁴

Y' = 4⋅(areing x)³ ⋅ ¹/_1+x²

Y' = 4(areing x)³ = ^{4⋅areing3(x)}/_1+x²

8

Y = 2 ⋅ arccos(x/2)

Y' = -1 / √(1 - (x/2)²) ⋅ (1/2)

Y' = -1 / √(1 - x²/4)

Y' = -1 / √(1/4) - x²

Y' = -1 / √(1 - x²)

Y' = -√(1/4) / √(1 - x²)

Y' = -2 / √(1 - x²)

Y = 2 \operatorname{arcsin} \sqrt{x}

Y' = \frac{2}{\sqrt{1-(\sqrt{x})^2}} \cdot \frac{1}{2\sqrt{x}}

Y' = \frac{2}{\sqrt{1-x}} \cdot \frac{1}{2\sqrt{x}}

Y' = \frac{1}{\sqrt{x}(1-x)}

N. B. \quad \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}

Y = ln(\operatorname{arcsin} x)

N. B. \quad y = ln(f \circ g(x))

Y' = x' \cdot \frac{f'(g(x)) \cdot g'(x)}{f(g(x))}

Y' = \frac{1}{\operatorname{arcsin} x} \cdot \frac{1}{\sqrt{1-x^2}}

Y' = \frac{1}{\operatorname{arcsin} x \sqrt{1-x^2}}

Y = \operatorname{arcsin}^{2} x

Y' = 2 \operatorname{arcsin} x \cdot \frac{1}{\sqrt{1-x^2}}

Y' = \frac{2 \operatorname{arcsin} x}{\sqrt{1-x^2}}

M^o 482

Y = arccos (e^x+1)

Y' = -¹/_{√1+(e^x+1)²} · e^x

Y' = -^ex/_1+(e^x+1)² ;

M^o 483

Y = arcsin ^x/₂

Y' = ¹/_√1(^x/2)² · ¹/₂

Y' = ¹/_√1-^x2/4 ;

Y' = ¹/_√^1-x2/4 ;

Y' = ¹/_√1-^x2/2 · ¹/₂

Y' = ¹/_√1-^x2/4

Y' = ¹/_{2 √1-x² ;}

Y = arccos (cos ^x/₂)

Y' = ¹/_{√1-sin² x / ^x/2} · ^{cos x}/₂

Y' = ¹/_{√cos²x / ^x/2} · ¹/₂

Y' = ¹/_{cos ^x/2} · ^{cos x}/₂ · ¹/₂

Y' = ¹/₂

Dx determin. per ux e costante !!

Y = arccos (cos x)

Y' = -1 / √(1 - cos² x) (-sen x)

Y' = -1 / √sen² x (- sen x)

Y' = - 1 / |sen x| (- sen x)

Y' = + 1

Y' = - 1

Se sen x > 0

Se sen x

Y = x arctan (2x)

Y' = 1. arctan(2x) + x . 1 / 1 + (2x)² . 2

Y = arctan(2x) + 2x / 1 + 4x²

Y' = -1 / √1-(1-x²)² · (-2x)

Y' = 2x / √1-(1-2x²+x⁴)

Y' = 2x / √1 + 2x² - x⁴

Y' = 2x / √x²(2-x²)

Y' = 2x / |x|·√2-x²

N.b. √x² = |x|

Y = 2, arcceos

Y' = ²/_{1+^{(x2-1)/(x2+1)2}} · ^{x(x2+1-(x2-1)(2x))}/_(x²+1)²

Y' = ²/_{1+^{(x2-1)/(x2+1)2}} - ^{x(1/2+x2/(x2+1))}/_(x²+1)²

Y' = ²/_{(x²+1)²+((x²-1)²/(x²+1)²)} - ^2x·x/_(x²+1)²

Y' = - ^2(x/(x2+1))2/_{(x²+1)+((x²-1)²)} - ^2x/_(x²+1)²

Y' = - ^8x/_{x²+1+2x²+xⁿ-2x²+1}

Y = - ^8x/_2xⁿ+2 = - ^8x/_x(xⁿ+1)

Y = - ^7x/_x⁴+1 ;

Y = arc\log ^x-1/_x+1 + arc\log ^x

Y'₁ = ¹/_{(x+1)² + (x-1)²/(x+1)²} . ^{1(x+1) - (x-1)(1)}/_(x+1)² + ¹/_1+x²

Y'₁ = ¹/_(x+1)² . ^{x+1 - x+1}/_(x+1)² + ¹/_1+x²

Y'₁ = ^(x+1)2/_{x²+2x+1+x²-2x+1} . ⁰/_(x+1)² + ¹/_1+x²

Y' = ²/_2(x+1) + ¹/_1+x²

Y' = ²/_x²+1

Y = arcsin(2x) + √(1−h x²)

Y' = ¹/_{√(1−[2x]²)} · 2x + ¹/_{√1−h x²} = (^-1 ¹/_{√1−h x²}

Y' = ²/_{√1−h x²} + ⁴/_{√1−h x²}

Y' = ^{2−h x}/_{√1−h x²} = ^2(1+2x)/_{√1−h x²};

Y = 1+lnx ∧ x>0

• A. Y = x + ¹/_x → Y' = 1 − ¹/_x² NO
• B. Y = x− ¹/_x → Y' = 1 + ¹/_x² NO
• C. Y = x + ln x → Y' = 1 + ¹/_x NO
• D. Y = x · ln x → Y' = 1 · ln x + x ¹/_x

Y = ln x + 1

Y' = 1 + ln x SI