Question

For which x does the derivative exist for the function f(x) = ln|sin(x)|?

a) x = 0
b) x = π/2
c) x = π
d) x = 3π/2

Answer 1

The derivative of the function f(x) = ln|sin(x)| exists for x = π/2 and x = 3π/2, where the sine function is nonzero.

Step-by-step explanation:

The derivative of the function f(x) = ln|sin(x)| exists at points where the inside function, sin(x), is defined and its absolute value is not equal to zero. This is because the logarithm function is undefined at zero, and the absolute value function can make the derivative undefined if the inside function transitions through zero.

• x = 0: Here, sin(0) = 0, so f(x) is undefined.
• x = π/2: Here, sin(π/2) = 1, and f(x) is defined and differentiable.
• x = π: Here, sin(π) = 0, so f(x) is undefined.
• x = 3π/2: Here, sin(3π/2) = -1, and f(x) is defined and differentiable.

Therefore, the derivative exists for x = π/2 and x = 3π/2.