I require to uncover $fracddx(frac1sqrtx+2)$ only using the basic an interpretation of the derivative $f"(x)=lim_h o 0fracf(x+h)-f(x)h$, but I"m not having any luck with the algebra. Any kind of hints?


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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 $h o0$.

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

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

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

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

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

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

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


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Ian MillerIan müller
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