1.
1) Фиксируем x. y(x) = 4x - 5
2) Даём приращение аргументу. y(x + Δx) = 4(x + Δx) - 5 = 4x + 4Δx - 5
3) находим приращение функции. Δy = y(x + Δx) - y(x) = 4x + 4Δx - 5 - 4x + 5 = 4Δx
4) Составляем отношение Δy/Δx. Δy/Δx = 4Δx/Δx = 4
5) Находим предел:
f'(x) = 4
2.
f(x) = x³ - 3x² + 5x + 3
f'(x) = 3x² - 3*2x + 5 + 0 = 3x² - 6x + 5
f'(-1) = 3*(-1)² + 6 + 5 = 3 + 11 = 14
3.
f(x) = eˣ * cosx
f'(x) = eˣ * cosx + eˣ * (-sinx) = eˣ (cosx - sinx)
f'(0) = e⁰(cos0 - sin0) = 1 * (1 - 0) = 1
4.
f(x) = (x²+2)/(x-3)
f'(x) = (2x(x-3) - (x²+2)) / (x-3)² = (2x² - 6x - x² - 2) / (x-3)² = (x²-6x-2) / (x-3)²
f'(4) = (4² - 24 - 2) / (4 - 3)² = 16 - 26 = -10
5.
f(x) = x^(1/4)
f'(x) = 1/4 * x^(1/4-1) = 1/4 * x^(-3/4)
f'(16) = 1/4 * 16^(-3/4) = 1/4 * 2⁻³ = 1/4 * 1/8 = 1/32
1) Фиксируем x. y(x) = 4x - 5
2) Даём приращение аргументу. y(x + Δx) = 4(x + Δx) - 5 = 4x + 4Δx - 5
3) находим приращение функции. Δy = y(x + Δx) - y(x) = 4x + 4Δx - 5 - 4x + 5 = 4Δx
4) Составляем отношение Δy/Δx. Δy/Δx = 4Δx/Δx = 4
5) Находим предел:
f'(x) = 4
2.
f(x) = x³ - 3x² + 5x + 3
f'(x) = 3x² - 3*2x + 5 + 0 = 3x² - 6x + 5
f'(-1) = 3*(-1)² + 6 + 5 = 3 + 11 = 14
3.
f(x) = eˣ * cosx
f'(x) = eˣ * cosx + eˣ * (-sinx) = eˣ (cosx - sinx)
f'(0) = e⁰(cos0 - sin0) = 1 * (1 - 0) = 1
4.
f(x) = (x²+2)/(x-3)
f'(x) = (2x(x-3) - (x²+2)) / (x-3)² = (2x² - 6x - x² - 2) / (x-3)² = (x²-6x-2) / (x-3)²
f'(4) = (4² - 24 - 2) / (4 - 3)² = 16 - 26 = -10
5.
f(x) = x^(1/4)
f'(x) = 1/4 * x^(1/4-1) = 1/4 * x^(-3/4)
f'(16) = 1/4 * 16^(-3/4) = 1/4 * 2⁻³ = 1/4 * 1/8 = 1/32