Derivatives of Trigonometric Functions

        Formulas for finding the derivative of the trigonometric functions are given. We assume that the trigonometric functions are functions of real numbers (angles measured in radians) because the trigonometric differentiation formulas rely on limit formulas that become more complicated if the degree measurement is used instead of radian measure.

Proposition (Derivatives of the Trigonometric Functions) The trigonometric functions sine, cosine, tangent, cotangent, cosecant, and secant are all differentiable functions on their domain and their derivative functions are:

        

        

        

    Proof. For the derivative of the cosine function, we use the formula



along with the definition of the derivative:















For the derivative of the sine function, we use the formula



along with the definition of the derivative:















For the derivative of the tangent function, we use the formula along with the quotient rule:



For the derivative of the cotangent function, we use the formula   along with the quotient rule:



For the derivative of the secant function, we use the formula   along with the quotient rule:



For the derivative of the cosecant function, we use the formula   along with the quotient rule:

    Since the trigonometric functions are differentiable functions on their domains they are also continuous functions on their domain.

Example (Derivatives of the Trigonometric Functions) Find the derivative functions for the functions and

    Solution. For the function we use the quotient rule, derivative rules for sine and cosine, and a few trigonometric identites, we determine,
    




and simplifies to,   For the function we use the quotient rule and the derivative rules for sine and cosine, we determine,

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