If the equation F(x,y,z) = 0 determines z as a...

If the equation F(x,y,z) = 0 determines z as a differentiable function of x and y, then, at the points where Fz ? 0, the following equations are true. \frac{dz}{dx} = -\frac{F_x}{F_z} and \frac{dz}{dy} = -\frac{F_y}{F_z} Use these equations to find the values of dz/dx and dz/dy at the given point z^3 - 5xy + 5yz + 2y^3 - 155 = 0, (3, 4, 3) \frac{dz}{dx}|_{(3,4,3)} = \boxed{} (Type an integer or a simplified fraction) \frac{dz}{dy}|_{(3,4,3)} = \boxed{} (Type an integer or a simplified fraction)

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00:01 The given question we have two functions f and g z z f is that f is function of x y z and w this is equal to zero and g of x y z and w this is also equal to zero and thereby z and w are differentiable functions of independent variables x and y so that's in case one since x and y are independent we can find df upon d x then x and y are already dependent y is is x and y are independent so this can be written as d f by d x into d x by d x by d x by d x by d x s d f by t y into t y by d x x d f by d z into d z by d x x x d f by d w into d w by d w by d w by d d f so this will be equal to one so d f by d x plus d f by d y into d y by d x is 0 as df by d z into d z x x plus d f by d w into d w by d x since f is 0 so this will be equal to 0 so from this equation we can write d f by d w into d w into d w by d w by d x is equal to minus d f by d x minus d f by d x minus d f by d z into t z z by d x so thereby d w by d x will be equal to minus d f by d x minus d f by d z into d z z by d f by d w this equation one so similarly d g by d x when x and y are independent can be equal to zero from here will get d g by d x into x x x plus d g by d y to d y by d y by d x into plus d g by d z into d x plus d g by d w into d w by d x so similar to f here also this term will be 0 and this term will be 1 by simplifying and this will be equal to 0 so simplifying we'll get d w by d x will be x will be equal to minus dj by d x minus d g by d z into d z x over upon d g by d w so this is equation two so equating equation one and two get minus d f by these x minus d f by d z into x minus d f by d z into d x over upon d f by d w will be equal to minus d g by d x minus d x minus d j by d x to d x to over upon d d w so this will be when cross multiplying this we get minus d f by d x into t j by d w minus t f by d z into t z by d x into t z by d x by d x into t z by d x to to dc by d double is equal to minus d f by d .w into d .c by dx minus d g by d z into d z by dx into d f into d z by dx into t f in to d f f t w...

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