Решить примеры на фото.

Разберем каждое уравнение и найдем производную функции y.

Вариант 1

1) (y = 8x - 34)

(y' = 8)

2) (y = 6x^2 - 3x + 12)

(y' = 12x - 3)

3) (y = 3x^9 - 4x^6 + 5x^3 - 6x + 3)

(y' = 27x^8 - 24x^5 + 15x^2 - 6)

4) (y = 25 \(\cot\)(x))

(y' = -25 \(\csc\)^2(x)) или (y' = -\(\frac{25}{\sin^2(x)}\))

5) \(y = \frac{1}{7}x^{\frac{4}{5}} - 2\)

\(y' = \frac{1}{7} \cdot \frac{4}{5}x^{\frac{4}{5} - 1} = \frac{4}{35}x^{-\frac{1}{5}} = \frac{4}{35\sqrt[5]{x}}\)

6) \(y = \frac{1}{2}x^{14} - 2x\)

(y' = 7x^{13} - 2)

7) (y = \(\log\)_6(x) - 6x)

(y' = \(\frac{1}{x \ln(6)}\) - 6)

8) (y = \(\cos\)(x) \(\cdot\) x)

(y' = -x\(\sin\)(x) + \(\cos\)(x))

Вариант 2

1) (y = 3x - 81)

(y' = 3)

2) (y = 7x^2 - 4x + 123)

(y' = 14x - 4)

3) (y = 4x^{12} - 2x^9 + 5x^2 - 8x + 32)

(y' = 48x^{11} - 18x^8 + 10x - 8)

4) (y = 27 \(\tan\)(x))

(y' = 27 \(\sec\)^2(x)) или (y' = \(\frac{27}{\cos^2(x)}\))

5) \(y = \frac{1}{9}x^9\)

(y' = x^8)

6) \(y = \frac{1}{4}x^{12}\)

(y' = 3x^{11})

7) (y = \(\log\)_1(x))

Производная не определена, так как логарифм по основанию 1 не существует.

8) (y = \(\tan\)(x) \(\cdot\) x)

(y' = x\(\sec\)^2(x) + \(\tan\)(x))

Вариант 3

1) (y = 6x - 13)

(y' = 6)

2) (y = 4x^2 - 5x + 10)

(y' = 8x - 5)

3) (y = 2x^{10} - 3x^5 + 4x^2 - 5x + 2)

(y' = 20x^9 - 15x^4 + 8x - 5)

4) (y = 23 \(\sin\)(x))

(y' = 23 \(\cos\)(x))

5) \(y = \frac{1}{5}x^5\)

(y' = x^4)

6) \(y = \frac{1}{3}x^{12} - 5x\)

(y' = 4x^{11} - 5)

7) (y = \(\log\)_4(x))

(y' = \(\frac{1}{x \ln(4)}\))

8) (y = \(\cos\)(x) \(\cdot\) x)

(y' = \(\cos\)(x) - x \(\sin\)(x))