Given f(x) = x/sqrt(1-x^2)" [1]"f(x)=x√1−x2 [1]
f(x+h)= (x+h)/sqrt(1-(x+h)^2)" [2]"f(x+h)=x+h√1−(x+h)2 [2]
f'(x) = lim_(hto0) (f(x+h)-f(x))/h" [3]"
Substitute equation [1] and [2] into equation [3]:
f'(x) = lim_(hto0) ((x+h)/sqrt(1-(x+h)^2)-x/sqrt(1-x^2))/h" [3.1]"
Make a common denominator by multiplying the numerator and denominator by 1 in the form (sqrt(1-(x+h)^2)sqrt(1-x^2))/(sqrt(1-(x+h)^2)sqrt(1-x^2)):
f'(x) = lim_(hto0) ((x+h)sqrt(1-x^2)-xsqrt(1-(x+h)^2))/(hsqrt(1-(x+h)^2)sqrt(1-x^2))" [3.2]"
Multiply numerator and denominator by 1 in the form ((x+h)sqrt(1-x^2)+xsqrt(1-(x+h)^2))/((x+h)sqrt(1-x^2)+xsqrt(1-(x+h)^2)) this will make the numerator become the difference of two squares:
f'(x) = lim_(hto0) ((x+h)^2(1-x^2)-x^2(1-(x+h)^2))/(hsqrt(1-(x+h)^2)sqrt(1-x^2)((x+h)sqrt(1-x^2)+xsqrt(1-(x+h)^2)))" [3.3]"
Simplify the numerator (a lot):
f'(x) = lim_(hto0) (h^2+2hx)/(hsqrt(1-(x+h)^2)sqrt(1-x^2)((x+h)sqrt(1-x^2)+xsqrt(1-(x+h)^2)))" [3.4]"
There is common factor of h/h to 1:
f'(x) = lim_(hto0) (h+2x)/(sqrt(1-(x+h)^2)sqrt(1-x^2)((x+h)sqrt(1-x^2)+xsqrt(1-(x+h)^2)))" [3.5]"
We may, now, let h to 0 without any problems:
f'(x) = (2x)/(sqrt(1-(x)^2)sqrt(1-x^2)((x)sqrt(1-x^2)+xsqrt(1-(x)^2)))" [3.6]"
Simplify the denominator:
f'(x) = (2x)/((2x)(1-(x)^2)sqrt(1-x^2))" [3.7]"
(2x)/(2x) to 1 and the denominator becomes the 3/2 power:
f'(x) = 1/(1-x^2)^(3/2)" [3.8]"
