Question

Найти f’(-1), если f(x)=4x^3-2/x^2-1/x^3

Answer 1

Решение во вложениии

Answer 2

To find the derivative of the function f(x) and then evaluate it at x = -1, we'll follow these steps:

Step 1: Find the derivative of f(x) with respect to x. Step 2: Substitute x = -1 into the derivative to find f'(-1).

Let's start with Step 1:

Given the function f(x) = 4x^3 - 2/x^2 - 1/x^3, we'll first rewrite it to make the notation clearer:

f(x) = 4x^3 - 2x^(-2) - x^(-3)

Now, let's find the derivative f'(x):

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

To find the derivatives of each term, we use the power rule:

d/dx (x^n) = n * x^(n-1)

So, applying the power rule to each term:

f'(x) = 3 * 4x^(3-1) - (-2) * 2x^(-2-1) - (-3) * x^(-3-1) f'(x) = 12x^2 + 4x^(-3) + 3x^(-4)

Now, let's proceed to Step 2:

We need to evaluate f'(-1), which means substituting x = -1 into the expression we found for f'(x):

f'(-1) = 12(-1)^2 + 4(-1)^(-3) + 3(-1)^(-4) f'(-1) = 12(1) + 4(-1)^(-3) + 3(-1)^(-4) f'(-1) = 12 - 4 + 3 f'(-1) = 11

Therefore, the value of f'(-1) is 11.