Question

F(x)=(3arcsinX+7arctgX)'

Answer 1

Производная = 3 \√(1-х²) + 7 \ (1+х²)

Answer 2

To find the derivative of the function f(x) = (3arcsin(x) + 7arctan(x)), we'll use the chain rule and the derivative formulas for arcsin(x) and arctan(x).

The derivative of arcsin(x) is given by:

d/dx arcsin(x) = 1 / sqrt(1 - x^2)

And the derivative of arctan(x) is given by:

d/dx arctan(x) = 1 / (1 + x^2)

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

f(x) = 3arcsin(x) + 7arctan(x)

f'(x) = (3arcsin(x))' + (7arctan(x))'

Applying the chain rule to each term:

f'(x) = 3 * (1 / sqrt(1 - x^2)) * (arcsin(x))' + 7 * (1 / (1 + x^2)) * (arctan(x))'

Now, we need to find the derivatives of arcsin(x) and arctan(x), which are:

(arcsin(x))' = 1 / sqrt(1 - x^2)

(arctan(x))' = 1 / (1 + x^2)

Substituting these derivatives back into the expression for f'(x), we get:

f'(x) = 3 * (1 / sqrt(1 - x^2)) * (1 / sqrt(1 - x^2)) + 7 * (1 / (1 + x^2)) * (1 / (1 + x^2))

Simplifying further:

f'(x) = 3 / (1 - x^2) + 7 / ((1 + x^2)^2)

Therefore, the derivative of the function f(x) = (3arcsin(x) + 7arctan(x)) is:

f'(x) = 3 / (1 - x^2) + 7 / ((1 + x^2)^2)