Question

Compute the derivative of the given function.
f(x)=2 sin x/3 cos x-6

fʹ(x)=___

Answer 1

The derivative of f(x) = (2 sin x)/(3 cos x - 6) is found using the quotient rule and is f'(x) = (6 cos^2 x + 6 sin^2 x - 12 cos x)/(9 cos^2 x - 36 cos x + 36).

Step-by-step explanation:

To compute the derivative of the function f(x) = (2 sin x)/(3 cos x - 6), you need to apply the quotient rule since the function is a ratio of two functions. The quotient rule states that if you have a function h(x) = u(x)/v(x), then its derivative h'(x) = (v(x)u'(x) - u(x)v'(x))/(v(x))^2.

Let's define u(x) = 2 sin x and v(x) = 3 cos x - 6. Then calculate the derivatives of u and v with respect to x, which are u'(x) = 2 cos x and v'(x) = -3 sin x. Now apply the quotient rule:

f'(x) = ((3 cos x - 6)(2 cos x) - (2 sin x)(-3 sin x))/((3 cos x - 6)^2).

Simplifying the equation, you get:

f'(x) = (6 cos^2 x + 6 sin^2 x - 12 cos x)/(9 cos^2 x - 36 cos x + 36).