Page:Calculus Made Easy.pdf/191

 So we have this curious result that we have found a function such that if we differentiate it twice over, we get the same thing from which we started, but with the sign changed from $$+$$ to $$-$$.

The same thing is true for the cosine; for differentiating $$\cos\theta$$ gives us $$-\sin\theta$$, and differentiating $$-\sin\theta$$ gives us $$-\cos\theta$$; or thus:

Sines and cosines are the only functions of which the second differential coefficient is equal (and of opposite sign to) the original function.

Examples.

With what we have so far learned we can now differentiate expressions of a more complex nature.

(1) $$y=\arcsin x$$.

If $$y$$ is the arc whose sine is $$x$$, then $$x=\sin y$$.

Passing now from the inverse function to the original one, we get

a rather unexpected result.