Question

Y=6^sinxнайти производную функции.

Answer 1

Сначало находим внешнюю функцию g(x) = 6^sinx * ln6
затем находим внутреннюю f(x) = cosx
тогда получаем y= 6^sinx * ln6 * cosx  

Answer 2

Finding the Derivative of Y = 6^sin(x)

To find the derivative of the function Y = 6^sin(x), we can use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

The derivative of Y with respect to x, denoted as dY/dx, can be found using the chain rule as follows:

Step 1: Identify the outer function and the inner function. - The outer function is the exponentiation function, f(u) = 6^u, where u = sin(x). - The inner function is u = sin(x).

Step 2: Find the derivative of the outer function with respect to u. - The derivative of the outer function f(u) = 6^u with respect to u is f'(u) = ln(6) * 6^u.

Step 3: Find the derivative of the inner function with respect to x. - The derivative of the inner function u = sin(x) with respect to x is u' = cos(x).

Step 4: Apply the chain rule to find the derivative of Y with respect to x. - Using the chain rule, the derivative of Y = 6^sin(x) with respect to x is: dY/dx = f'(u) * u' dY/dx = ln(6) * 6^sin(x) * cos(x).

Therefore, the derivative of Y = 6^sin(x) with respect to x is dY/dx = ln(6) * 6^sin(x) * cos(x).