MALE: Find the directional derivative D sub u of f of x, y of f of x, y equal to 3x squared times y, minus 4y cubed, plus 3y squared, minus 4x in the direction of the vector u equal to cosine, parentheses, pi over 3, close parentheses i, plus sine, open parentheses, pi over 3, close parentheses, j. What is D sub u of f evaluated at 3,4.

By definition, D sub u of f of x,y is equal to fx of xy, multiplied by cosine theta, plus f sub y of x,y, multiplied by sine of theta.

Now, we know the angle theta because our vector is defined in terms of cosines and sines. So the angle pi over 3 is theta. So let me just first note that. Along with that, we need to find the partials of that function. So our f of xy, f sub x is equal to 6xy, minus 4y cubed, minus 4. And f sub y is equal to 3x squared, minus 12xy squared, plus 6y.

All right, now because we also know our angle is pi over 3, we need to find cosine of pi over 3. That is, I'll just say cosine of theta. Cosine theta is equal to 1/2, and sine theta is equal to the square root of 3 over 2.

Putting that together, D sub u of f of x,y is equal to, open parentheses, 6xy, minus 4y cubed, minus 4, close parentheses, multiplied by 1/2, that is r cosine of theta, plus, open parentheses, 3x squared, minus 12xy squared, plus 6y, close parentheses, multiply the square root of 3 over 2.

Evaluating this expression at the point 3,4, our directional derivative in the direction of u of this function, f of xy, at the point 3,4 is negative 94, minus a fraction of the numerator of 525, square root of 3, and a denominator of 2.