Question

Which expression represents StartFraction d y Over d x EndFraction if y = ln(3 − 3cos(3x))?

StartFraction 1 Over 3 minus 3 cosine (3 x) EndFraction
StartFraction sine (3 x) Over 1 minus cosine (3 x) EndFraction
StartFraction sine (3 x) Over 3 minus 3 cosine (3 x) EndFraction
StartFraction 3 sine (3 x) Over 1 minus cosine (3 x)

Answer 1

The expression that represents dy/dx is sine(3x) / (3 - 3cos(3x)).

Step-by-step explanation:

The expression that represents dy/dx is sine(3x) / (3 - 3cos(3x)).

In this case, y = ln(3 - 3cos(3x)).

To find dy/dx, we will use the Chain Rule. The Chain Rule states that if we have a composition of functions, such as f(g(x)), then the derivative of that composition is given by f'(g(x)) * g'(x).

In our case, the outer function is y = ln(u) and the inner function is u = 3 - 3cos(3x). We need to find the derivatives of both y and u to apply the Chain Rule.

The derivative of y = ln(u) with respect to u is 1/u.

The derivative of u = 3 - 3cos(3x) with respect to x is 3sin(3x).

Now, we can apply the Chain Rule: dy/dx = (1/u) * (du/dx).

Substituting the values, we have dy/dx = (1/(3 - 3cos(3x))) * (3sin(3x)).