in a physics exercise I need the derivative of . I don't want not only look in my Bronstein for the derivative, I want to calculate it on my own by using the theorem of the inverse function. The tangent is in the intervall strict monoton so that it is bijective and the inverse exists.
Now I calculate:
Now I have the problem, that I don't know how to express this in for a clear correspondence. I looked in my mathbook and found the relation:
With this it is clear, because according to assumption and I got:
Could someone explain me the relation please? I thought about trigonometric relations like pythagoras but I did not see it.
Thanks for help
Just in case a picture helps...
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
Follow clockwise from top left.
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!