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

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Understanding the Derivative of sin(1/x)

The function in question is sin(1/x). To find its derivative, we will use the chain rule of differentiation, which is a fundamental technique in calculus.

Step-by-Step Derivation

The chain rule states that if you have a composite function f(g(x)), the derivative is given by:

f'(g(x)) * g'(x)

In this case, let:

  • f(u) = sin(u) where u = 1/x
  • g(x) = 1/x

Calculating the Derivatives

Now we need to calculate the derivatives of f(u) and g(x):

  1. Derivative of f(u):

    Using the derivative of sine:

    f'(u) = cos(u)

    Thus, f'(g(x)) = cos(1/x)

  2. Derivative of g(x):

    For g(x) = 1/x, we use the power rule:

    g'(x) = -1/x^2

Applying the Chain Rule

Now that we have both derivatives, we can apply the chain rule:

dy/dx = f'(g(x)) * g'(x) = cos(1/x) * (-1/x^2)

Conclusion

The derivative of sin(1/x) is:

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

This result implies that the rate of change of the function sin(1/x) depends both on the cosine of the reciprocal of x and the square of x.

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