Total differential to calculate approximately the largest error

[Ereisorhet]

I have the following problem:
Use the total differential to calculate approximately the largest error at determine the area of a triangle rectangle (right triangle) from the lengths of the cathetus if they measure 6 and 8 cm respectively, with a possible error of 0.1 cm for each measurement.

I did this:
View attachment 9034

 

[I like Serena]

I have the following problem:
Use the total differential to calculate approximately the largest error at determine the area of a triangle rectangle (right triangle) from the lengths of the cathetus if they measure 6 and 8 cm respectively, with a possible error of 0.1 cm for each measurement.

I did this:

Greetings hero, and welcome to MHB!

Your end result is correct.

What you wrote to get there, is not quite right though. For starters there is no actual reference to the total differential.
Let me clean it up a bit.

Let $A$ be the area of the triangle, and let $a$ and $b$ be the lengths of the sides.
The total differential is then $dA$.
And:
\begin{aligned}
A&=\frac 12 ab \\
dA&=d\left(\frac 12 ab\right) = \frac 12 b\,da + \frac 12 a\,db \\
\text{largest error} &\approx \left| \frac 12 b\,da \right| + \left|\frac 12 a\,db\right| = \left| \frac 12\cdot 6\cdot 0.1 \right| + \left| \frac 12\cdot 8\cdot 0.1 \right| = 0.3 + 0.4 = 0.7
\end{aligned}