Rate of change problem with velocities

Question

I can't seem to get this problem right.

Can someone help me with the formula for $$ \frac {df}{dt} $$

Answer 1

At $\theta =\pi/3$,

$$ f=\sqrt{x^2+y^2-xy}$$

Take the time derivatives $f’=f_x’x’+f_y’y’$,

$$f’ = \frac12 \frac {(2x-y)x’ + (2y-x)y’}{\sqrt{x^2+y^2-xy}}$$

Then, plug in $x=28$, $y=12$, $x’=-2$ and $y’=-5$ to obtain

$$f’ = \frac12 \frac {(56-12)\cdot (-2)+ (24-28)\cdot (-5)}{\sqrt{28^2+12^2-28\cdot 12}}= -\frac{17\sqrt{37}}{74}$$