Derivatives of trigonometric functions from "summary" of Differential Calculus by S Balachandra Rao
The derivatives of trigonometric functions play a crucial role in the study of calculus. These derivatives help us understand the rate of change of trigonometric functions at any given point. By finding the derivatives of trigonometric functions, we can determine the slope of the tangent line to the curve represented by these functions. The derivatives of the basic trigonometric functions - sine, cosine, and tangent - have specific formulas that can be derived using the definitions of these functions and the rules of differentiation. For example, the derivative of the sine function, denoted as d(sin x)/dx, is equal to the cosine function, i. e., d(sin x)/dx = cos x. Similarly, the derivative of the cosine function is equal to the negative of the sine function, i. e., d(cos x)/dx = -sin x. In addition to the basic trigonometric functions, we can also find the derivatives of other trigonometric functions such as secant, cosecant, and cotangent. These derivatives can be derived using the fundamental definitions of these functions and the chain rule of differentiation. For instance, the derivative of the secant function is equal to the secant function multiplied by the tangent function, i. e., d(sec x)/dx = sec x * tan x. It is essential to understand the derivatives of trigonometric functions to solve various problems in calculus, such as finding maximum and minimum values of trigonometric functions, determining the concavity of curves, and analyzing the behavior of functions at critical points. By mastering the derivatives of trigonometric functions, one can enhance their problem-solving skills and gain a deeper insight into the behavior of trigonometric functions in calculus.
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