Question

Вычислить: f'(2) и f'(4),если f(x) = 2х'2 +х'3.​

Answer 1

Ответ:

Пошаговое объяснение:

$\[\begin{gathered}f(x)=2{x^2}+{x^3}\hfill \\f'(x)={\left( {2{x^2}+{x^3}} \right)^\prime}=4x+3{x^2}\hfill \\f'(2)=4x+3{x^2}=4\cdot2+3\cdot{2^2}=8+12=20\hfill \\f'(4)=4x+3{x^2}=4\cdot 4+3\cdot{4^2}=16+48=64\hfill \\ \end{gathered} \]$

Answer 2

To find the derivative of the function f(x) = 2x^2 + x^3, we'll use the power rule for differentiation, which states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Let's find the first derivative of f(x):

f(x) = 2x^2 + x^3

Using the power rule, we differentiate each term with respect to x:

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

Now, we can find the value of the derivative at x = 2 and x = 4:

1. f'(2): f'(2) = 4 * 2 + 3 * 2^2 f'(2) = 8 + 12 f'(2) = 20

2. f'(4): f'(4) = 4 * 4 + 3 * 4^2 f'(4) = 16 + 48 f'(4) = 64

Therefore, f'(2) = 20 and f'(4) = 64.