Найдите значение производной функции f(x) = x^5 * sin 7x + 2х в точке x= 0

To solve the problem, we need to find the derivative of the function \( f(x) = x^5 \cdot \sin(7x) + 2x \) and then evaluate this derivative at the point \( x = 0 \).

### Step 1: Differentiate the function
The function is:
\[
f(x) = x^5 \cdot \sin(7x) + 2x
\]

We'll apply the product rule to the first term and differentiate the second term directly.

1. Differentiate \( x^5 \cdot \sin(7x) \) using the product rule:
   \[
   \text{If } u(x) = x^5 \text{ and } v(x) = \sin(7x), \text{ then } \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
   \]
   \[
   u'(x) = 5x^4, \quad v'(x) = 7\cos(7x)
   \]
   \[
   \frac{d}{dx}[x^5 \cdot \sin(7x)] = 5x^4 \cdot \sin(7x) + x^5 \cdot 7\cos(7x)
   \]

2. Differentiate \( 2x \):
   \[
   \frac{d}{dx}[2x] = 2
   \]

So, the derivative of \( f(x) \) is:
\[
f'(x) = 5x^4 \cdot \sin(7x) + 7x^5 \cdot \cos(7x) + 2
\]

### Step 2: Evaluate the derivative at \( x = 0 \)
Substituting \( x = 0 \) into the derivative:
\[
f'(0) = 5(0)^4 \cdot \sin(7 \cdot 0) + 7(0)^5 \cdot \cos(7 \cdot 0) + 2
\]
\[
f'(0) = 0 + 0 + 2 = 2
\]

So, the value of the derivative at \( x = 0 \) is \( \boxed{2} \).

Would you like further details or have any questions? Here are some related questions you might find useful:

1. What is the product rule in differentiation?
2. How do you differentiate trigonometric functions?
3. What is the derivative of a polynomial function?
4. How does the chain rule apply to trigonometric functions?
5. How do you evaluate a derivative at a specific point?
6. What are the common mistakes when applying the product rule?
7. How do you handle higher-order derivatives?
8. What is the importance of the derivative in calculus?

**Tip:** When differentiating products of functions, always apply the product rule carefully to avoid mistakes.

Learn how to find the derivative of the function f(x) = x^5 * sin(7x) + 2x at x = 0 with a detailed step-by-step solution. This problem involves differentiation using the product rule and differentiation of trigonometric functions. Suitable for advanced high school students and calculus enthusiasts.

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