Question

Найдите производную (tgx)^(6x)

Answer 1

Finding the Derivative of (tgx)^(6x)

To find the derivative of the function (tgx)^(6x), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative of f(g(x)) is given by f'(g(x)) * g'(x).

In this case, our composite function is (tgx)^(6x), where f(u) = u^6 and g(x) = tgx. Let's find the derivatives of f(u) and g(x) separately and then apply the chain rule.

The derivative of f(u) = u^6 is given by f'(u) = 6u^5.

The derivative of g(x) = tgx is given by g'(x) = sec^2(x).

Now, let's apply the chain rule:

f'(g(x)) = f'(tgx) * g'(x)

Substituting the derivatives we found earlier:

f'(tgx) = 6(tgx)^5

g'(x) = sec^2(x)

Therefore, the derivative of (tgx)^(6x) is:

(6(tgx)^5) * sec^2(x)

Note: Please keep in mind that the above answer is based on the information provided in the search results. It's always a good practice to double-check the calculations and consult reliable sources for confirmation.