given a simple function from trigonometry = Sin (x) is differentiable at every point of the entire domain.It is necessary to prove that the derivative of the sine of any argument is the cosine of the same angle, that is, '= Cos (x).
proof is based on the definition of derivative
Define x (arbitrary) in a small neighborhood of a particular point of △ x x0.We show the value of a function in it, and at the point x to find the increment of the specified function.If △ x - increment of the argument, then a new argument - is x0 + Δx = x, the value of this function at a given value of the argument y (x) is Sin (x0 + Δx), the value of a function at a specific point at (x0) is also known.
Now we have Δu = Sin (X0 + △ x) -Sin (x0) - received the increment function.
According to the formula of sine sum of two unequal angles will convert the difference Δu.
Δu = Sin (x0) · Cos (△ x) + Cos (x0) · Sin (Δx) minus Sin (x0) = (Cos (Δx) -1) · Sin (x0) + Cos (x0) · Sin (△ x).
swapping of terms grouped the first to the third Sin (x0), carried a common factor - sine - the brackets.We got to express the difference Cos (△ x) -1.You're changing the sign of the bracket and in parentheses.Knowing what is the 1-Cos (△ x), we make the change and obtain a simplified expression Δu, which is then divided by △ x.
Δu / △ x is of the form: Cos (x0) · Sin (△ x) / △ x 2 · Sin2 (0,5 · △ x) · Sin (x0) / △ x.This is the ratio of the increment function to assumptions increment argument.
remains to find the limit of the ratios obtained by us during lim △ x tends to zero.
known that limit Sin (△ x) / Δx is equal to 1, for a given condition.And the expression 2 · Sin2 (0,5 · △ x) / △ x of the resulting sum the private transformation to a product containing as first remarkable limit factor: the numerator of the fraction and znemenatel divide by 2, the square of the sine replace the product.So:
(Sin (0,5 · Δx) / (0,5 · Δx)) · Sin (Δx / 2).
limit of this expression as △ x tends to zero, the number is equal to zero (1 multiplied by 0).It turns out that the limit of the ratio Δy / △ x is equal to Cos (x0) · 1-0, this is Cos (x0), an expression that does not depend on △ x, tending to 0. Hence the conclusion: the derivative of the sine of any angle x is equal to cosine of xwe write so: '= Cos (x).
This formula is listed in the table of known derivatives, where all elementary functions
When solving problems, where he meets the derivative of the sine, you can use the rules of differentiation and ready-made formulas from the table.For example, to find the derivative of a simple function y = 3 · Sin (x) -15.We use the basic rules of differentiation, the removal of the numerical factor for the sign of the derivative and derivative computation constant number (it is zero).Apply the tabulated value of the derivative of the sine of the angle x equal Cos (x).We get the answer: y '= 3 · Cos (x) -O.This derivative, in turn, is also an elementary function y = G · Cos (x).
derivative of the sine squared of any argument
When calculating the expression (Sin2 (x)), you need to remember how to differentiate a complex function.So, = Sin2 (x) - is an exponential function as sine squared.The argument it is also a trigonometric function, a complicated argument.The result in this case is the product of the first factor is the derivative of the square of a complex argument, and the second - a derivative of the sinus.Here is the rule for differentiating a function of a function: (u (v (x))) 'is (u (v (x)))' · (v (x)) '.Expression v (x) - a complex argument (internal function).If the given function is "y is equal to sine squared x", the derivative of a composite function is y = 2 · Sin (x) · Cos (x).The product of the first factor is doubled - known derivative of a power function, and Cos (x) - derivative of the sine of the argument of complex quadratic function.The final result can be converted by using the formula of the trigonometric sine of the double angle.A: The derivative is Sin (2 · x).This formula is easy to remember, it is often used as a table.
