Tính đạo hàm

To compute the derivatives for the functions listed, we can apply standard calculus rules such as the power rule, product rule, quotient rule, chain rule, and logarithmic differentiation. Here's a summary of the derivatives for each function:

1. **a)** \( y = 3x^2 - \frac{2}{x} + 3\sqrt{x} \)

\( y' = 6x + \frac{2}{x^2} + \frac{3}{2\sqrt{x}} \)

2. **b)** \( y = \frac{5x - 1}{2x + 3} \)

Using the quotient rule:
\( y' = \frac{(5)(2x + 3) - (5x - 1)(2)}{(2x + 3)^2} = \frac{(10x + 15) - (10x - 2)}{(2x + 3)^2} = \frac{17}{(2x + 3)^2} \)

3. **c)** \( y = 6^x - 5\ln{x} - \sqrt{2} \)

\( y' = 6^x\ln{6} - \frac{5}{x} \)

4. **d)** \( y = \sqrt{e^x \sin{2x}} \)

Using the chain rule and product rule:
\( y' = \frac{1}{2\sqrt{e^x \sin{2x}}} \left( e^x\sin{2x} + e^x(2\cos{2x}) \right) = \frac{e^x(\sin{2x} + 2\cos{2x})}{2\sqrt{e^x \sin{2x}}} \)

5. **e)** \( y = 5 - \log_3(7x^2 - x) \)

\( y' = -\frac{1}{(7x^2 - x)\ln{3}}(14x - 1) \)

6. **f)** \( y = (3 - x^2)\sqrt{x} \)

Using the product rule:
\( y' = \sqrt{x}(-2x) + (3 - x^2)\frac{1}{2\sqrt{x}} = \frac{-2x\sqrt{x} + (3 - x^2)}{2\sqrt{x}} \)

7. **g)** \( y = \cos^4(5x) \)

Using the chain rule:
\( y' = 4\cos^3(5x)(-5\sin(5x)) = -20\cos^3(5x)\sin(5x) \)

8. **h)** \( y = (1 + \tan{x})^3 \)

Using the chain rule:
\( y' = 3(1 + \tan{x})^2(\sec^2{x}) \)

9. **i)** \( y = (\cot{x} - 3)^4 \)

Using the chain rule:
\( y' = 4(\cot{x} - 3)^3(-\csc^2{x}) \)

These derivatives represent the rates of change of the respective functions with respect to \( x \).