By elementary school algebra (try it!) friend can present that$$fracf(x+h)-f(x)h=-frac1sqrtx+h+2,sqrtx+2,,(sqrtx+h+2+sqrtx+2)$$and that is then easy to discover the limit together $h o0$.

You are watching: Derivative of square root of x 1

$$fracddx frac1sqrtx+2=lim_h o0fracleft(frac1sqrtx+h+2-frac1sqrtx+2 ight)h$$

$$=lim_h o0fracleft(frac1sqrtx+h+2-frac1sqrtx+2 ight)h imesfracfrac1sqrtx+h+2+frac1sqrtx+2frac1sqrtx+h+2+frac1sqrtx+2$$

$$=lim_h o0fracleft(frac1x+h+2-frac1x+2 ight)hleft(frac1sqrtx+h+2+frac1sqrtx+2 ight)$$

$$=lim_h o0fracleft(frac-h(x+h+2)(x+2) ight)hleft(frac1sqrtx+h+2+frac1sqrtx+2 ight)$$

$$=lim_h o0fracleft(frac-1(x+h+2)(x+2) ight)left(frac1sqrtx+h+2+frac1sqrtx+2 ight)$$

$$=fracfrac-1(x+2)^2frac2sqrtx+2$$

$$=frac-12(x+2)^frac32$$

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answer Jun 2 "16 in ~ 4:48

Ian MillerIan müller

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