Question

Найти производную функции: 1) f(x)=2x^3 2)h(x)=3/x 3)f(x)=x^4-4x^3-8x^2+13 4)v(x)= -1/h+3h 5)f(x)=2/3x+3x^2 6)f(x)=2x^7/7 - 3x^5/10 +4x^3 - x - 10 7)f(x)= - x^5/5 + 2x^3/3 - x^2 + x + 5/x^2

Answer 1

Решение прилагается к картинке

Answer 2

To find the derivatives of the given functions, we'll use the rules of differentiation. The general rule for finding the derivative of a function f(x) with respect to x is denoted as f'(x) or dy/dx. Here are the derivatives for each function:

1. f(x) = 2x^3 f'(x) = d/dx (2x^3) = 2 * 3x^(3-1) = 6x^2

2. h(x) = 3/x h'(x) = d/dx (3/x) = -3/x^2

3. f(x) = x^4 - 4x^3 - 8x^2 + 13 f'(x) = d/dx (x^4) - d/dx (4x^3) - d/dx (8x^2) + d/dx (13) = 4x^3 - 4 * 3x^2 - 8 * 2x + 0 = 4x^3 - 12x^2 - 16x

4. v(x) = -1/h + 3h v'(x) = d/dx (-1/h) + d/dx (3h) = -(-1/x^2) + 3 * (d/dx(x)) = 1/x^2 + 3

5. f(x) = 2/3x + 3x^2 f'(x) = d/dx (2/3x) + d/dx (3x^2) = 2/3 * d/dx(x) + 3 * 2x^(2-1) = 2/3 + 6x

6. f(x) = 2x^7/7 - 3x^5/10 + 4x^3 - x - 10 f'(x) = d/dx (2x^7/7) - d/dx (3x^5/10) + d/dx (4x^3) - d/dx (x) - d/dx (10) = 2/7 * d/dx (x^7) - 3/10 * d/dx (x^5) + 4 * 3x^(3-1) - 1 - 0 = 2/7 * 7x^6 - 3/10 * 5x^4 + 12x^2 - 1 = 2x^6 - 3/2 x^4 + 12x^2 - 1

7. f(x) = -x^5/5 + 2x^3/3 - x^2 + x + 5/x^2 f'(x) = d/dx (-x^5/5) + d/dx (2x^3/3) - d/dx (x^2) + d/dx (x) + d/dx (5/x^2) = -1/5 * d/dx (x^5) + 2/3 * d/dx (x^3) - d/dx (x^2) + 1 - 5 * d/dx (x^-2) = -1/5 * 5x^4 + 2/3 * 3x^2 - 2x + 1 + 5 * 2x^-3 = -x^4 + 2x^2 - 2x + 1 + 10/x^3

These are the derivatives of the given functions with respect to x.