$\sqrt{4{\left(3.03\right)}^3\+5{\left(6.2\right)}^2}$

$\doteq f\left(3\,6\right)\+\mathrm{df}$

$\=\sqrt{4\cdot 27\+5\cdot 36}\+\left(f_x\left(3\,6\right)\cdot \left(.03\right)\+f_y\left(3\,6\right)\cdot \left(.2\right)\right)$

$\=\sqrt{276}\+\left(\frac{9}{4}\sqrt{2}\cdot \left(.03\right)\+\frac{5}{4}\sqrt{2}\cdot \left(.2\right)\right)$

$\=16.61324773\+0.4490128060$

$\=17.06226054$

Define $f\left(x\,y\right)$ and its first partial derivatives

$f\left(x\,y\right)\=\sqrt{4 x^3\+5 y^2}$$\stackrel{\text{assign as function}}{\to }$$f$

$\mathrm{f\_\_x}\left(x\,y\right)\=\frac{\partial }{\partial x} f\left(x\,y\right)$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_x}$$$

$\mathrm{f\_\_y}\left(x\,y\right)\=\frac{\partial }{\partial y} f\left(x\,y\right)$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_y}$

Calculate $f\left(3.03\,6.2\right)\doteq f\left(3\,6\right)\+\mathrm{df}$ 

$f\left(3\,6\right)\+\left(\mathrm{f\_\_x}\left(3\,6\right)\cdot \left(.03\right)\+\mathrm{f\_\_y}\left(3\,6\right)\cdot \left(.2\right)\right)$ = $1.026458333\sqrt{288}$$\stackrel{\text{at 10 digits}}{\to }$$17.41957555$$$

Compute $f\left(3.03\,6.2\right)$ "exactly"

$f\left(3.03\,6.2\right)$ = $17.42046234$$$

Define an appropriate function $f$ 

$f≔\left(x\,y\right)\to \sqrt{4 x^3\+5 y^2}\:$

Compute $\mathrm{df}\=f_x\left(3\,6\right)\cdot \left(0.03\right)\+f_y\left(3\,6\right)\cdot \left(0.2\right)$ 

$\mathrm{df}≔\mathrm{evalf}\left({\mathrm{D}}_1\left(f\right)\left(3\,6\right)\cdot \left(.03\right)\+{\mathrm{D}}_2\left(f\right)\left(3\,6\right)\cdot \left(.2\right)\right)$ = $0.4490128060$$$

Approximate $f\left(3.03\,6.2\right)$ as $f\left(3\,6\right)\+\mathrm{df}$ 

$\mathrm{evalf}\left(f\left(3\,6\right)\+\mathrm{df}\right)$ = $17.41957555$$$

Obtain $f\left(3.03\,6.2\right)$ "exactly"

$f\left(3.03\,6.2\right)$ = $17.42046234$$$