2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

The derivatives of tangent, cotangent, secant, and cosecant functions are important concepts in Calculus that allow us to find the rate of change of these trigonometric functions. These derivatives can be found using the basic trigonometric identities and the chain rule. The derivative of the tangent function (tan) is given by the formula: (tan(x))' = sec^2(x) The derivative of the cotangent function (cot) is given by the formula: (cot(x))' = -csc^2(x) The derivative of the secant function (sec) is given by the formula: (sec(x))' = sec(x)tan(x) The derivative of the cosecant function (csc) is given by the formula: (csc(x))' = -csc(x)cot(x) It's important to note that these derivatives are only valid for angles in radians, not degrees.

Example 1: Find the derivative of f(x) = tan(2x) Solution: Using the derivative formula for tangent, the derivative of f(x) = tan(2x) = sec^2(2x) = (1/cos^2(2x)) = (1/cos^2(2x)) Example 2: Find the derivative of f(x) = cot(3x) Solution: Using the derivative formula for cotangent, the derivative of f(x) = cot(3x) = -csc^2(3x) = -(1/sin^2(3x)) Example 3: Find the derivative of f(x) = sec(4x) Solution: Using the derivative formula for secant, the derivative of f(x) = sec(4x) = sec(4x)tan(4x) = (1/cos(4x))tan(4x) Example 4: Find the derivative of f(x) = csc(5x) Solution: Using the derivative formula for cosecant, the derivative of f(x) = csc(5x) = -csc(5x)cot(5x) = -(1/sin(5x))cot(5x) Example 5: Find the derivative of f(x) = tan^2(6x) Solution: Using the derivative formula for tangent and the chain rule, the derivative of f(x) = tan^2(6x) = 2tan(6x)sec^2(6x) = 2tan(6x)(1/cos^2(6x)) Example 6: Find the derivative of f(x) = cot^3(x) Solution: Using the derivative formula for cotangent, the chain rule and the power rule, the derivative of f(x) = cot^3(x) = -3cot^2(x)csc^2(x) = -3cot^2(x)(1/sin^2(x)) Example 7: Find the derivative of f(x) = sec^4(x) Solution: Using the derivative formula for secant and the chain rule, the derivative of f(x) = sec ^4(x) = 4sec^3(x)tan(x) = 4(1/cos^3(x))tan(x)

In conclusion, finding the derivatives of tangent, cotangent, secant, and cosecant functions is an important concept in Calculus. These derivatives can be found using the basic trigonometric identities and the chain rule. It's important to keep in mind that these derivatives are only valid for angles in radians, not degrees. It's also important to practice and understand the use of chain rule and power rule when differentiating complex trigonometric functions.