In details

Derivatives of inverse trigonometric functions


A common problem in trigonometry is finding an angle whose trig functions are known.

Problems of this kind involve the computation of bow functionssuch as arcsen xarccos xarctg x, and so on. Consider this idea from the point of view of inverse functions, with the goal of developing derivative formulas for inverse trigonometric functions.

Identities for inverse trigonometric functions

If we interpret x as an angle measured in radians whose sine is x, and if that angle is not negative, then we can represent x as an angle in a right triangle, where the hypotenuse has length 1 and the opposite side to the angle of has length x (figure a). By the Pythagorean Theorem, the adjacent side to the angle has length .

In addition, the angle opposite to é , since the cosine of that angle is x (figure b). This triangle motivates several useful identities, involving trigonometric functions that are valid for . For example:

Similarly, x and x can be represented with angles of right triangles shown in figure c and d. These triangles reveal more useful identities, such as:


NOTE. Nothing is gained by memorizing these identities; what is important is to understand the method used to get them.

Example

The figure below shows a computer generated graph of y = (sen x). You might think this graph should be the straight line y = x, once (sen x) = x. Why does not this happen?

Solution. The relationship (sen x) = x is valid in range ; soon we can say for sure that the graphs of y = (sen x) and y = x coincide in this range. However, outside this range, the relationship (sen x) = x It does not have to be valid. For example, if you are in the range then the amount x - will be in range . Like this

Thus, using the identity sen (x-) = -sen x and the fact that it's an odd function we can express (sen x) how

This shows that in the range , the graph of y = (sen x) coincides with the line y = -(x-), which has slope -1 and an intercept x in x = , which is in accordance with The figure.

Derivation formula

Remember that if f is a one-to-one function whose derivative is known, so there are two basic ways to get a derivation formula for (x), we can rewrite the equation y = (x) how x = f(y), and differentiate implicitly. We will use implicit differentiation to obtain the derivation formula for y = x. Rewriting this equation as x = sen y and differentiating implicitly we get

This derivative formula can be simplified by applying the formula , which was deduced from the triangle of the figure, resulting:

Thus, we show that

If u is a differentiable function of x, then and the chain rule produce the following generalized derivative formula

The method used to obtain this formula can also be used to obtain generalized derivative formulas for other inverse trig functions. These formulas, valid for -1 < u <1, are

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