By primary school algebra (try it!) girlfriend can show that$$fracf(x+h)-f(x)h=-frac1sqrtx+h+2,sqrtx+2,,(sqrtx+h+2+sqrtx+2)$$and the is then simple to find the limit as $h o0$.
You are watching: Derivative of sqrt(1-x^2)
$$fracddx frac1sqrtx+2=lim_h o0fracleft(frac1sqrtx+h+2-frac1sqrtx+2 ight)h$$
$$=lim_h o0fracleft(frac1sqrtx+h+2-frac1sqrtx+2 ight)h imesfracfrac1sqrtx+h+2+frac1sqrtx+2frac1sqrtx+h+2+frac1sqrtx+2$$
$$=lim_h o0fracleft(frac1x+h+2-frac1x+2 ight)hleft(frac1sqrtx+h+2+frac1sqrtx+2 ight)$$
$$=lim_h o0fracleft(frac-h(x+h+2)(x+2) ight)hleft(frac1sqrtx+h+2+frac1sqrtx+2 ight)$$
$$=lim_h o0fracleft(frac-1(x+h+2)(x+2) ight)left(frac1sqrtx+h+2+frac1sqrtx+2 ight)$$
answer Jun 2 "16 in ~ 4:48
Ian MillerIan müller
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