Differentiation is used to prove that arcsin(x) + arccos(x) = ?/2.

Let f(x) = arcsin(x) + arccos(x)We first prove that f(x) is a constant function. First find the derivative of f.

f '(x) = d( arcsin(x) )/dx + d( arccos(x) )/dx

= 1 / sqrt(1 - x

= 0

Now if f '(x) = 0 for all values of x, then that means that f(x) is a constant function that may be calculated using any value of x. Let us use x = 0 and x = 1.(Note that one value is enough).

f(0) = arcsin(0) + arccos(0) = 0 +?/2 =?/2

f(1) = arcsin(1) + arccos(1) =?/2 + 0 =?/2

Hence arcsin(x) + arccos(x) =?/2 for all values of x.

More on applications of differentiation

applications of differentiation