This example calculates 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 right-hand side of the equation.
Simplify the factorial operator in the denominator.
Now the right-hand side is very close-to, 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 .
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Derive Cosine Function (Taylor Series) Example
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Derive Sine Function (Taylor Series) Example