What is the derivative of sin(1/x)?

To find the derivative of the function sin(1/x), we will use the chain rule of differentiation.

Let y = sin(1/x). The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) imes g'(x).

In this case, we have:

• f(u) = sin(u) where u = 1/x
• So, f'(u) = cos(u)
• And, g(x) = 1/x where g'(x) = -1/x^2

Putting this together, we apply the chain rule:

dy/dx = cos(1/x) imes (-1/x^2).

Thus, the derivative of sin(1/x) is:

-cos(1/x) / (x^2).

This gives us the slope of the function sin(1/x) at any point x not equal to zero (since x=0 is undefined for this function).