Explications étape par étape :
EX1
f(x) = -x² f'(x) = -2x
f(x) = 18√x = 18 × x¹/² f'(x) = 9x¹/² ⁻¹ = 9x⁻¹/² = 9 × 1/x¹/² = 9/√x
f(x) = -3/x f'(x) = -3x⁻¹ = 3x⁻² = 3/x²
f(x) = 4x⁴ - 5x³ - 3x² - 7 f'(x) = 16x³ - 15x² - 6x
f(x) = ( 2x² - x + 1 ) ( -7x + 8 ) y = U × V y' = U'V + UV'
f'(x) = ( 4x - 1 ) ( -7x + 8 ) + ( 2x² - x + 1 ) × (-7)
⇔ f'(x) = ( 4x - 1 ) ( -7x + 8 ) -7 ( 2x² - x + 1 )
⇔ f'(x) = -28x² + 32x + 7x - 8 - 14x² + 7x - 7
⇔ f'(x) = -42x² + 46x - 15
f(x) = ( x² + 3x - 7 ) / ( x + 5 ) y = U/V y' = ( U'V - UV' ) / V²
f'(x) = [ ( 2x+ 3 ) ( x + 5 ) - ( x² + 3x - 7 ) ] / ( x + 5 )²
⇔ f'(x) = ( 2x² + 10x + 3x + 15 - x² - 3x + 7 ) / ( x + 5 )²
⇔ f'(x) = ( x² + 10x + 22 ) / ( x + 5 )²
EX2
f(x) = x² + 3x - 1
f'(x) = 2x + 3
1) Equation de la tangente: y = f'(a) ( x - a ) + f(a)
f(2) = 2² + 3 × 2 - 1 = 9
f'(2) = 2 × 2 + 3 = 7
y = f'(2) ( x - 2 ) + f(2)
y = 7 ( x - 2 ) + 9
⇔ y = 7x - 14 + 9
⇔ y = 7x - 5
2) f(x) - y(x)
x² + 3x - 1 - ( 7x - 5 )
x² + 3x - 1 - 7x + 5
x² - 4x + 4
( x - 2 )²
x - ∞ 2 + ∞
f(x) - y(x) + +
f(x) ≥ y(x) sur ] - ∞ ; + ∞ [
Cf au-dessus de Cy
En espérant t'avoir aidé ... et que tu aies compris .
