This example finds of the derivative of the sine function using its taylor series definition.
Start with the taylor series definition of the sine function^{[1]}.
Take the derivative of both sides of the equation.
From the summation property of derivatives we can apply the power rule to each of the expressions on the righthand 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 righthand side.
The derivative of is .

Derive Sine Function (Taylor Series)Wumbo (internal)

Derive Cosine Function (Taylor Series)Wumbo (internal)