Differentiation of inverse trigonometric functions is a small and specialized topic. However, these particular derivatives are interesting to us for two reasons. First, computation of these derivatives provides a good workout in the use of the chain rul e, the definition of inverse functions, and some basic trigonometry. Second, it turns out that the derivatives of the inverse trigonometric functions are actually algebraic functions!! This is an unexpected and interesting connection between two seeming ly very different classes of functions.
It is possible to form inverse functions for restricted versions of all six basic trigonometric functions. One can construct and use an inverse cosecant function, for example. However, it is generally enought to consider the inverse sine and the inverse tangent functions. We will restrict our attention to these two functions.
Here are the results.


In order to verify the differentiation formula for the arcsine function, let us set
y = arcsin(x).We want to compute dy/dx. The first step is to use the fact that the arcsine function is the inverse of the sine function. Among other things, this means that
sin(y) = sin(arcsin(x)) = x.Next, differentiate both ends of this formula. We apply the chain rule to the left end, remembering that the derivative of the sine function is the cosine function and that y is a differentiable function of x.

= 



=  1 

= 

=  sec(y) 
If sin(y)=x, what is cos(y) in terms of x?
1^{2}=x^{2}+The last step is to express the trigonometric functions of y in terms of ratios of side lengths in the reference triangle. In particular, the secant of y is equal to the hypotenuse length divided by the adjacent side length.(third side)^{2}
(third side)^{2} = 1x^{2}
third side =sqrt(1x^{2})

= 

=  sec(y)  = 

This completes our study of differentiation for now. In Stage 6, we will investigate another general differentiation technique called implicit differentiation. Later still, we will learn how to differentiate exponential and logarithmic functions . For now, you should go to the Practice area and spend some time learning to use the many differentiation techniques that have been introduced in this Lesson. If there is one skill that we must develop for success in differential calculus, it is di fferentiation!! Enjoy, and good luck!!
If you find that you are having difficulty with differentiation, don't worry. You're not the first person to struggle with this technical skill. Contact your classmates. Discuss your difficulties. Contact your instructor. We can learn to do this!!
We now turn to the exponential and logarithmic functions. We have already discussed "the" exponential function: exp(x)=e^{x}. In order to differentiate the "other" exponential functions and the logarithmic functions, we must first compute the derivative of the inverse to the exponential function. Thus we turn our attention to the natural logarithm.
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William A. Bogley
Robby Robson