237.1) 2⁴−³+3+4; 2) −⁵+2³−3²−1; 3) 6∛+1/²; 4) 2/³−8∜; 5) (2+3)⁸; 6) (4−3)⁷; 7) ∛3−2; 8) 1/√1−4; 9) sin0,5; 10) cos(−3).

1) 2⁴−³+3+4

Производная: \[(2x^4 - x^3 + 3x + 4)' = 2 \cdot 4x^3 - 3x^2 + 3 = 8x^3 - 3x^2 + 3\]

2) −⁵+2³−3²−1

Производная: \[(-x^5 + 2x^3 - 3x^2 - 1)' = -5x^4 + 6x^2 - 6x\]

3) 6∛+1/² = 6x^(1/3) + x^(-2)

Производная: \[\left(6x^{\frac{1}{3}} + x^{-2}\right)' = 6 \cdot \frac{1}{3}x^{-\frac{2}{3}} - 2x^{-3} = 2x^{-\frac{2}{3}} - 2x^{-3} = \frac{2}{\sqrt[3]{x^2}} - \frac{2}{x^3}\]

4) 2/³−8∜ = 2x^(-3) - 8x^(1/4)

Производная: \[\left(2x^{-3} - 8x^{\frac{1}{4}}\right)' = -6x^{-4} - 8 \cdot \frac{1}{4}x^{-\frac{3}{4}} = -6x^{-4} - 2x^{-\frac{3}{4}} = -\frac{6}{x^4} - \frac{2}{\sqrt[4]{x^3}}\]

5) (2+3)⁸

Производная: \([(2x+3)^8]' = 8(2x+3)^7 \cdot 2 = 16(2x+3)^7\]

6) (4−3)⁷

Производная: \([(4-3x)^7]' = 7(4-3x)^6 \cdot (-3) = -21(4-3x)^6\]

7) ∛3−2 = (3x-2)^(1/3)

Производная: \([\sqrt[3]{3x-2}]' = \frac{1}{3}(3x-2)^{-\frac{2}{3}} \cdot 3 = (3x-2)^{-\frac{2}{3}} = \frac{1}{\sqrt[3]{(3x-2)^2}}\]

8) 1/√1−4 = (1-4x)^(-1/2)

Производная: \[(\frac{1}{\sqrt{1-4x}})' = \left((1-4x)^{-\frac{1}{2}}\right)' = -\frac{1}{2}(1-4x)^{-\frac{3}{2}} \cdot (-4) = 2(1-4x)^{-\frac{3}{2}} = \frac{2}{\sqrt{(1-4x)^3}}\]

9) sin0,5

Производная: \([\[\sin(0.5x)\]]' = 0.5 \cos(0.5x)\]

10) cos(−3)

Производная: \([\[\cos(-3x)\]]' = -\sin(-3x) \cdot (-3) = 3 \sin(-3x)\]