$f\left(2.03\,3.12\,1.48\right)$

$\doteq f\left(2\,3\,3\/2\right)\+\mathrm{df}$

$\=2\ln \left(\frac{9}{17}\right)\+\left(f_x\left(2\,3\/2\right)\cdot \left(.03\right)\+f_y\left(2\,3\,3\/2\right)\cdot \left(.12\right)\+f_z\left(2\,3\,3\/2\right)\cdot \left(-\mathrm{.}02\right)\right)$

$\=2\ln \left(\frac{9}{17}\right)\+\left(\left(\ln \left(\frac{9}{17}\right)-\frac{16}{17}\right)\cdot \left(.03\right)\+\frac{14}{153}\cdot \left(.12\right)-\frac{40}{153}\cdot \left(-\mathrm{.}02\right)\right)$

$\= -1.271977533-0.03110580679$

$\= -1.303083340$

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

$f\left(x\,y\,z\right)\=x \ln \left(\frac{y\+ z}{x^2\+y z}\right)$$\stackrel{\text{assign as function}}{\to }$$f$

Calculate $f\left(2.03\,3.12\,1.48\right)\doteq f\left(2\,3\,3\/2\right)\+\mathrm{df}$ 

$f\left(2\,3\,3\/2\right)\+\left(\mathrm{f\_\_x}\left(2\,3\,3\/2\right)\cdot \left(.03\right)\+\mathrm{f\_\_y}\left(2\,3\,3\/2\right)\cdot \left(.12\right)\+\mathrm{f\_\_z}\left(2\,3\,3\/2\right)\cdot \left(-.02\right)\right)$ = $2.03\ln \left(\frac{9}{17}\right)-0.01202614379$$\stackrel{\text{at 10 digits}}{\to }$$-1.303083340$

Compute $f\left(2.03\,3.12\,1.48\right)$ "exactly"

$f\left(2.03\,3.12\,1.48\right)$ = $-1.302614961$$$

Define an appropriate function $f$ 

$f≔\left(x\,y\,z\right)\to x \ln \left(\frac{y\+ z}{x^2\+y z}\right)\:$

Compute $\mathrm{df}\=f_x\left(2\,3\,3\/2\right)\cdot \left(0.03\right)\+f_y\left(2\,3\,3\/2\right)\cdot \left(0.12\right)\+f_z\left(2\,3\,3\/2\right)\cdot \left(-0.02\right)$ 

$\mathrm{df}≔\mathrm{evalf}\left({\mathrm{D}}_1\left(f\right)\left(2\,3\,3\/2\right)\cdot \left(.03\right)\+{\mathrm{D}}_2\left(f\right)\left(2\,3\,3\/2\right)\cdot \left(.12\right)\+{\mathrm{D}}_3\left(f\right)\left(2\,3\,3\/2\right)\cdot \left(-.02\right)\right)$

$-0.03110580679$

Approximate $f\left(2.03\,3.12\,1.48\right)$ as $f\left(2\,3\,3\/2\right)\+\mathrm{df}$ 

$\mathrm{evalf}\left(f\left(2\,3\,3\/2\right)\+\mathrm{df}\right)$ = $-1.303083340$$$

Obtain $f\left(2.03\,3.02\,4.97\right)$ "exactly"

$f\left(2.03\,3.12\,1.48\right)$ = $-1.302614961$$$