The process for differentiating a trigonometric function involves using the rules of differentiation, such as the power rule or chain rule, to find the derivative of the function. This involves taking the derivative of each term in the function and simplifying the result.
The chain rule is a rule of differentiation that allows us to find the derivative of a composite function. When differentiating a trigonometric function, the chain rule is used to differentiate the inner function first, then multiply it by the derivative of the outer function.
The most commonly used trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Their derivatives are as follows:
- d/dx(sin x) = cos x
- d/dx(cos x) = -sin x
- d/dx(tan x) = sec^2 x
- d/dx(cot x) = -csc^2 x
- d/dx(sec x) = sec x * tan x
- d/dx(csc x) = -csc x * cot x
The derivatives of the inverse trigonometric functions are as follows:
- d/dx(arcsin x) = 1/sqrt(1-x^2)
- d/dx(arccos x) = -1/sqrt(1-x^2)
- d/dx(arctan x) = 1/(1+x^2)
- d/dx(arccot x) = -1/(1+x^2)
- d/dx(arcsec x) = 1/(x*sqrt(x^2-1))
- d/dx(arccsc x) = -1/(x*sqrt(x^2-1))
Some common trig identities used in differentiating trig functions include:
- sin^2 x + cos^2 x = 1
- tan x = sin x / cos x
- cot x = cos x / sin x
- sec x = 1 / cos x
- csc x = 1 / sin x
- d/dx(ln x) = 1/x