This example finds of the derivative of the cosine 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.
Now the righthand side is very closeto, but not quire the taylor series of the sine function^{[2]}; each term’s sign in the expression is flipped. To fix this we can factor from each of the expressions.
Substitute in for its taylor series.
The derivative of is .

Derive Cosine Function (Taylor Series)Wumbo (internal)

Derive Sine Function (Taylor Series)Wumbo (internal)