Question

Y=ctg^4 1/3^x) найти dy

Answer 1

Ответ:

Если правильно записала))

Пошаговое объяснение:

Answer 2

To find dy/dx for the given function Y = ctg^4(1/3^x), we'll differentiate Y with respect to x.

First, let's rewrite the function using standard trigonometric functions:

ctg(x) = 1/tan(x)

So, Y = (1/tan(1/3^x))^4

Now, let's find the derivative:

dy/dx = d/dx [(1/tan(1/3^x))^4]

To differentiate Y with respect to x, we can use the chain rule:

d/dx [f(g(x))^n] = n * f'(g(x)) * g'(x), where f(x) = 1/x and g(x) = tan(1/3^x).

First, find the derivative of f(x):

f'(x) = d/dx [1/x] = -1/x^2

Now, find the derivative of g(x):

g'(x) = d/dx [tan(1/3^x)]

To differentiate tan(u), where u is a function of x, we use the chain rule again:

d/dx [tan(u)] = sec^2(u) * du/dx

Here, u = 1/3^x:

du/dx = d/dx [1/3^x] = -(1/ln(3)) * 3^(-x) = -(1/ln(3)) * (1/3^x)

So, g'(x) = sec^2(1/3^x) * (-(1/ln(3)) * (1/3^x))

Now, put it all together to find dy/dx:

dy/dx = 4 * (1/tan(1/3^x))^3 * (-1/ln(3)) * (1/3^x) * sec^2(1/3^x)

Finally, you can simplify further if needed.