Question

Производимая неявных функций ln(√(x^2+y^2))/2=arctg(y/x) ln(x)+√(y/x)=2 y=x(1-ln(x)) y=arctg√(4x^2-1) y=ln(arcsin(3x))

Answer 1

Решение на фотографии

Answer 2

To analyze and understand the given implicit functions, we'll go through each one separately:

1. ln(√(x^2+y^2))/2 = arctg(y/x)

To simplify this equation, we can start by isolating the natural logarithm term:

ln(√(x^2+y^2)) = 2 * arctg(y/x)

Next, we'll take the exponential of both sides to remove the logarithm:

√(x^2+y^2) = e^(2 * arctg(y/x))

Since e^(2 * arctg(y/x)) can be simplified as (e^(arctg(y/x)))^2 and e^(arctg(y/x)) is equal to √(1 + (y/x)^2), we get:

√(x^2+y^2) = (√(1 + (y/x)^2))^2

Now, simplify the right-hand side:

√(x^2+y^2) = 1 + (y/x)^2

Square both sides to eliminate the square root:

x^2 + y^2 = 1 + (y/x)^2

Multiply both sides by x^2:

x^2(x^2 + y^2) = x^2 + (y^2/x^2)

Simplify:

x^4 + x^2y^2 = x^2 + y^2

Move all terms to one side:

x^4 + x^2y^2 - x^2 - y^2 = 0

Now, this equation is in a somewhat simplified form, but it is challenging to explicitly solve for y in terms of x.

2. ln(x) + √(y/x) = 2

To solve for y in terms of x, we can start by isolating the square root term:

√(y/x) = 2 - ln(x)

Next, we square both sides to eliminate the square root:

y/x = (2 - ln(x))^2

Now, multiply both sides by x to isolate y:

y = x * (2 - ln(x))^2

3. y = x(1 - ln(x))

This equation is already solved for y in terms of x. It represents a logarithmic function.

4. y = arctg(√(4x^2 - 1))

This equation represents an inverse tangent function. It's already solved for y in terms of x.

5. y = ln(arcsin(3x))

This equation is also already solved for y in terms of x. It represents the natural logarithm of the arcsine function.

Please note that in some cases, the solutions might be limited to specific ranges of x to ensure that the functions are well-defined. Implicit functions can be challenging to manipulate and solve analytically in general, so the provided forms may be the most convenient representations.