Prove the Derivative of Sine (Taylor Series)

This example shows how to calculate the derivative of the sine function using a Taylor series.


  1. Start with the taylor series definition of the sine function[1].

  2. Take the derivative of both sides of the equation.

  3. From the summation property of derivatives, we can apply the power rule to each of the expressions on the right-hand side of the equation.

    Simplify the factorial operator in the denominator.

    This gives us the taylor series definition of the cosine function[2] which we can substitute in for on the right-hand side.

    The derivative of is .


  1. Derive Sine Function (Taylor Series)
    Wumbo (internal)
  2. Derive Cosine Function (Taylor Series)
    Wumbo (internal)