Question

differentiate implicitly to find the first partial derivatives of w. the function is (x^{2}+y^{2}+z^{2}-8yw + 3w^{2}=6). step 1: let (f(x,y,z,w)=x^{2}+y^{2}+z^{2}-8yw + 3w^{2}-6). step 2: differentiate (f) with respect to (x), (y), (z), and (w) respectively. (f_x = 2x) (f_y=) (f_z=) (f_w=)

Explanation:

Step1: Recall partial - derivative rules

To find the partial derivative of a function with respect to a variable, we treat other variables as constants.

Step2: Find $F_y$

Given $F(x,y,z,w)=x^{2}+y^{2}+z^{2}-8yw + 3w^{2}-6$. When finding $F_y$, we treat $x$, $z$, and $w$ as constants. The derivative of $x^{2}$ with respect to $y$ is $0$ (since $x$ is treated as a constant), the derivative of $y^{2}$ with respect to $y$ is $2y$, the derivative of $z^{2}$ with respect to $y$ is $0$ (since $z$ is a constant), the derivative of $- 8yw$ with respect to $y$ is $-8w$ (using the power - rule for $y$: $\frac{d(ay)}{dy}=a$ where $a=-8w$), and the derivative of $3w^{2}$ and $-6$ with respect to $y$ is $0$ (since they are constants with respect to $y$). So $F_y = 2y-8w$.

Step3: Find $F_z$

When finding $F_z$, we treat $x$, $y$, and $w$ as constants. The derivative of $x^{2}$ with respect to $z$ is $0$, the derivative of $y^{2}$ with respect to $z$ is $0$, the derivative of $z^{2}$ with respect to $z$ is $2z$, the derivative of $-8yw$ with respect to $z$ is $0$, and the derivative of $3w^{2}$ and $-6$ with respect to $z$ is $0$. So $F_z=2z$.

Step4: Find $F_w$

When finding $F_w$, we treat $x$, $y$, and $z$ as constants. The derivative of $x^{2}$ with respect to $w$ is $0$, the derivative of $y^{2}$ with respect to $w$ is $0$, the derivative of $z^{2}$ with respect to $w$ is $0$, the derivative of $-8yw$ with respect to $w$ is $-8y$, and the derivative of $3w^{2}$ with respect to $w$ is $6w$, and the derivative of $-6$ with respect to $w$ is $0$. So $F_w=-8y + 6w$.

Answer:

$F_y=2y - 8w$, $F_z = 2z$, $F_w=-8y+6w$