The image you uploaded contains a math task in Russian, asking to differentiate various functions using derivative tables and differentiation rules. Here are the functions given in the image:

1. a) $y = x^3 - 5\sqrt{x} - \frac{2}{x^2} + 3x^{\frac{3}{2}}$
2. b) $y = (2x + 1) \sin(3x)$
3. c) $y = \frac{e^{2x}}{2x - 1}$

I will help differentiate these functions step by step. Let's start with the first one.

1. a) $y = x^3 - 5\sqrt{x} - \frac{2}{x^2} + 3x^{\frac{3}{2}}$

To differentiate this, we apply basic differentiation rules:

• The derivative of $x^n$ is $nx^{n-1}$.
• The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$.

Let me compute the derivative for this.

Differentiating each term:

• $\frac{d}{dx}(x^3) = 3x^2$
• $\frac{d}{dx}(-5\sqrt{x}) = -5 \times \frac{1}{2\sqrt{x}} = -\frac{5}{2\sqrt{x}}$
• $\frac{d}{dx}\left(-\frac{2}{x^2}\right) = -2 \times (-2x^{-3}) = \frac{4}{x^3}$
• $\frac{d}{dx}(3x^{\frac{3}{2}}) = 3 \times \frac{3}{2}x^{\frac{1}{2}} = \frac{9}{2}x^{\frac{1}{2}}$

Now, putting all the derivatives together:

$\frac{dy}{dx} = 3x^2 - \frac{5}{2\sqrt{x}} + \frac{4}{x^3} + \frac{9}{2}x^{\frac{1}{2}}$

1. b) $y = (2x + 1) \sin(3x)$

For this function, we need to apply the product rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ Here, $f(x) = 2x + 1$ and $g(x) = \sin(3x)$.

• $f'(x) = 2$
• $g'(x) = 3\cos(3x)$ (derivative of $\sin(3x)$ is $3\cos(3x)$)

Now applying the product rule: $\frac{dy}{dx} = 2\sin(3x) + (2x + 1) \cdot 3\cos(3x)$ So, the derivative is: $\frac{dy}{dx} = 2\sin(3x) + 3(2x + 1)\cos(3x)$

1. c) $y = \frac{e^{2x}}{2x - 1}$

For this function, we use the quotient rule: [ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} ] Here, $f(x) = e^{2x}$ and $g(x) = 2x - 1$.

• $f'(x) = 2e^{2x}$ (derivative of $e^{2x}$)
• $g'(x) = 2$

Now applying the quotient rule: $\frac{dy}{dx} = \frac{2e^{2x}(2x - 1) - e^{2x}(2)}{(2x - 1)^2}$ Simplifying: $\frac{dy}{dx} = \frac{e^{2x}(4x - 2 - 2)}{(2x - 1)^2} = \frac{e^{2x}(4x - 4)}{(2x - 1)^2}$ This simplifies to: $\frac{dy}{dx} = \frac{4e^{2x}(x - 1)}{(2x - 1)^2}$

Would you like more details on the steps, or do you have any other questions? Here are some related questions you may find helpful:

1. How do you apply the chain rule to differentiate more complex functions?
2. Can the product and quotient rules be used for functions involving trigonometric and exponential terms?
3. What is the general approach for differentiating rational functions?
4. How do you simplify expressions after differentiating to make them more manageable?
5. How can the second derivative be used to determine concavity and inflection points?

Tip: Always look for ways to factor or simplify terms before differentiating complex expressions to make the process easier.