I am new to calculus and cannot see the logic of the following question… Any feedback will be really appreciated!

The function $$f(x,y,z)$$ is differentiable at all points, and satisfies $$f(x,y,2x^2+y^2)=4x+5y$$.

Point P is defined as (1,2,6).

Unit vector $$\mathbf{u}=(\frac{2}{3},\frac{1}{3},\frac{2}{3})$$.

$$D_\mathbf{u}f(P)=8.$$

Find the gradient for f(x,y,z) at the given point. Give the sum of elements of the gradient you found.

Options:

1. 4
2. 4.5
3. 5
4. 5.5

I am lost at the very beginning: I understand that I need to calculate $$f_x,f_y,f_z$$ in order to calculate the gradient. $$f_x=4$$,$$f_y=5$$, but I am stuck at $$f_z$$. I have understood that z depends on x and y (but then I don't quite understand why it is included in the definition of the function $$f(x,y,z)$$, as $$z$$ is not independent of $$x$$ and $$y$$). It does not appear on the right side of $$f(x,y,2x^2+y^2)=4x+5y$$, so I cannot seem to find a way to find its derivative either.

• I don't think you can say $f_x = 4$ and $f_y = 5$. It says $f$ is equal to $4x+5y$ only when $z = 2x^2+y^2$. – Nick Apr 21 at 16:18
• Also, are you sure the problem says $D_uf(P) = 8$? Is it possible this is a typo, and it should be $\frac{8}{3}$? If it is indeed 8/3, I get the answer 5.5. If it is 8, I get the answer is not one of the 4 choices. – Nick Apr 21 at 16:30
• It definitely says 8, but I will check if it's a typo…! – dalta Apr 21 at 17:12
• @nick But in the meanwhile, could you tell me how you computed the gradient? – dalta Apr 21 at 17:13

Let's say $$g(x,y) = 2x^2+y^2$$. Then the problem says $$f(x,y,g(x,y)) = 4x+5y$$. Use the chain rule to differentiate both sides with respect to both $$x$$ and $$y$$ to get

$$f_x + f_z g_x = f_x + 4xf_z = 4$$

$$f_y + f_z g_y = f_y + 2yf_z = 5$$

Plugging in the point $$P = (1,2,6)$$, you get

$$f_x + 4f_z = 4$$

$$f_y + 4f_z = 5$$

The fact that $$D_uf(P) = 8$$ means that $$\frac{1}{3} \left( 2f_x + f_y + 2f_z \right) = 8$$. Or, multiplying by $$3$$, you get $$2f_x + f_y + 2f_z = 24$$. Now you have three equations in three unknowns. You can solve the linear system to get

$$f_x = 8.9$$ $$f_y = 9.4$$ $$f_z = -1.1$$

The reason I think there may be a typo, and it might be $$D_uf(P) = \frac{8}{3}$$ is that in that case, you instead get $$f_x = 2$$, $$f_y = 3$$, $$f_z = 0.5$$.