Найти производную функции (831-839). 831 1) e* + 1; 2) ex + x²; 3) e2x + 1; 4) e-3x + √x. 832 1) e2x + 1 + 2x3; 4) e1-x + x-³; 833 1) 2x + e*; 2) 3x - x-²; 3) e2x – x; 4) e3x + 2x²; 5) 3x² +2. 834 1) 0,5% + e3x; 2) 3* - e2x; 3) e2-x+√x; 4) e3-x+1. + x 2) e*--√x-1; 5) ex²; 3) 0,3x+2+; x 6) e2x3. x4 835 1) 2 ln x + 3*; 4) 3 x-³ - log3 x; 836 1) sin x + x²; 2) cos x - 1; 2) 3 ln x - 2x; 5) ln (x² - 2x); 3) cos x + e*; 4) sin x - 2*. 3) log2 x + 1; 2x 6) (3x² - 2) log3 x.

Решение:

• 831

1. \( (e^x + 1)' = e^x \)
2. \( (e^x + x^2)' = e^x + 2x \)
3. \( (e^{2x} + \frac{1}{x})' = 2e^{2x} - \frac{1}{x^2} \)
4. \( (e^{-3x} + \sqrt{x})' = -3e^{-3x} + \frac{1}{2\sqrt{x}} \)

• 832

1. \( (e^{2x+1} + 2x^3)' = 2e^{2x+1} + 6x^2 \)
2. \( (e^{\frac{1}{2}x - 1} - \sqrt{x-1})' = \frac{1}{2}e^{\frac{1}{2}x - 1} - \frac{1}{2\sqrt{x-1}} \)
3. \( (e^{0.3x+2} + \frac{1}{\sqrt{x}})' = 0.3e^{0.3x+2} - \frac{1}{2x^{\frac{3}{2}}} \)
4. \( (e^{1-x} + x^{-3})' = -e^{1-x} - 3x^{-4} \)
5. \( (e^{x^2})' = 2xe^{x^2} \)
6. \( (e^{2x^3})' = 6x^2e^{2x^3} \)

• 833

1. \( (2^x + e^x)' = 2^x \cdot ln(2) + e^x \)
2. \( (3^x - x^{-2})' = 3^x \cdot ln(3) + 2x^{-3} \)
3. \( (e^{2x} - x)' = 2e^{2x} - 1 \)
4. \( (e^{3x} + 2x^2)' = 3e^{3x} + 4x \)
5. \( (3x^2 + 2)' = 6x \)

• 834

1. \( (0.5^x + e^{3x})' = 0.5^x \cdot ln(0.5) + 3e^{3x} \)
2. \( (3^x - e^{2x})' = 3^x \cdot ln(3) - 2e^{2x} \)
3. \( (e^{2-x} + \sqrt[3]{x})' = -e^{2-x} + \frac{1}{3x^{\frac{2}{3}}} \)
4. \( (e^{3-x} + \frac{1}{x^4})' = -e^{3-x} - \frac{4}{x^5} \)

• 835

1. \( (2 \cdot ln(x) + 3^x)' = \frac{2}{x} + 3^x \cdot ln(3) \)
2. \( (3 \cdot ln(x) - 2x)' = \frac{3}{x} - 2 \)
3. \( (log_2(x) + \frac{1}{2x})' = \frac{1}{x \cdot ln(2)} - \frac{1}{2x^2} \)
4. \( (3x^{-3} - log_3(x))' = -9x^{-4} - \frac{1}{x \cdot ln(3)} \)
5. \( (ln(x^2 - 2x))' = \frac{2x - 2}{x^2 - 2x} \)
6. \( ((3x^2 - 2) \cdot log_3(x))' = 6x \cdot log_3(x) + (3x^2 - 2) \cdot \frac{1}{x \cdot ln(3)} \)

• 836

1. \( (sin(x) + x^2)' = cos(x) + 2x \)
2. \( (cos(x) - 1)' = -sin(x) \)
3. \( (cos(x) + e^x)' = -sin(x) + e^x \)
4. \( (sin(x) - 2^x)' = cos(x) - 2^x \cdot ln(2) \)