Question

Найдите значение f ' (π/12), если f(x)=(1+sinx)2.

Answer 1

Решение дано на фото.

Answer 2

To find the derivative of the function f(x) = (1 + sin(x))^2 with respect to x and then evaluate it at x = π/12, we'll follow these steps:

Step 1: Find the derivative of f(x) with respect to x. Step 2: Evaluate the derivative at x = π/12.

Let's start with Step 1:

Step 1: Find f'(x): To find the derivative of f(x), we'll use the chain rule since we have a composition of functions: g(x) = (1 + x)^2 and h(x) = sin(x).

Using the chain rule, the derivative of f(x) is:

f'(x) = 2 * (1 + sin(x)) * (1 + sin(x))' = 2 * (1 + sin(x)) * cos(x).

Step 2: Evaluate f' at x = π/12: Now, we need to evaluate the derivative at x = π/12:

f'(π/12) = 2 * (1 + sin(π/12)) * cos(π/12).

To proceed further, we need to calculate sin(π/12) and cos(π/12):

sin(π/12) ≈ 0.2588190451, cos(π/12) ≈ 0.9659258263.

Now, we can find f'(π/12):

f'(π/12) = 2 * (1 + 0.2588190451) * 0.9659258263 ≈ 2 * 1.2588190451 * 0.9659258263 ≈ 2.4270509839.

So, the value of f'(π/12) is approximately 2.4270509839.