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For example, the derivative of the sine function is written sin ′ (a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as ...
By continuing this process, if it exists, the th derivative is the derivative of the th derivative or the derivative of order . As has been discussed above , the generalization of derivative of a function f {\displaystyle f} may be denoted as f ( n ) {\displaystyle f^{(n)}} . [ 32 ]
If y = f(x 1, ..., x n) and all of the variables x 1, ..., x n depend on another variable t, then by the chain rule for partial derivatives, one has = = + + = + +. Heuristically, the chain rule for several variables can itself be understood by dividing through both sides of this equation by the infinitely small quantity dt.
The derivative of sine is cosine, and the derivative of cosine is negative sine: [16] = (), = (). Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself. [ 15 ]
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) {\textstyle \arctan(y,x)} .
The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives. At a discontinuity, the series will converge to the average of the right and left limits.
If units of degrees are intended, the degree sign must be explicitly shown (sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/ π)°, so that, for example, sin π = sin 180° when we take x = π.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.