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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 tan(x) = sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation.
The derivative of order zero of f is defined to be f itself and ... (In addition, the series for ln(1 − x) converges for x = −1, and the series for ln ...
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same.
This may be obtained from the partial fraction decomposition of given above, which is the logarithmic derivative of . [22] From this, it can be deduced also that cos z = ∏ n = 1 ∞ ( 1 − z 2 ( n − 1 / 2 ) 2 π 2 ) , z ∈ C . {\displaystyle \cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad ...
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
The derivative of ′ is the second derivative, denoted as ″ , and the derivative of ″ is the third derivative, denoted as ‴ . By continuing this process, if it exists, the n {\displaystyle n} th derivative is the derivative of the ( n − 1 ) {\displaystyle (n-1)} th derivative or the derivative of order ...
(The series indicates that −J 1 (x) is the derivative of J 0 (x), much like −sin x is the derivative of cos x; more generally, the derivative of J n (x) can be expressed in terms of J n ± 1 (x) by the identities below.)
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.