Question

Differentiate implicitly to find the first partial derivatives of \(w\). \(x^{2}+y^{2}+z^{2}-5 y w+10 w^{2}=2\)

Short answer

\(\frac{\partial w}{\partial x} = 0\), \(\frac{\partial w}{\partial y} = \frac{2y-5w}{5y}\), and \(\frac{\partial w}{\partial z} = 0\).

Step-by-step solution

Differentiate Implicitly with respect to \(x\)

Differentiating the given equation with respect to \(x\), treating all other variables as constants, gives: \(2x + 0 + 0 - 0 - 0 = 0\), hence, \(\frac{\partial w}{\partial x} = 0\) as we don't have \(w\) in terms of \(x\) in this equation.

Differentiate Implicitly with respect to \(y\)

Differentiate the given equation with respect to \(y\), treating all other variables as constants. We have:\n\(0+2y-5w-5y\frac{\partial w}{\partial y}+0=0\) \nThis rearranges to \(5y\frac{\partial w}{\partial y} = 2y-5w\). \nTherefore, \(\frac{\partial w}{\partial y} = \frac{2y-5w}{5y}\).

Differentiate Implicitly with respect to \(z\)

Next, differentiate the given equation with respect to \(z\), again treating all other variables as constants. This gives: \(0+0+2z-0-0=0\), which simplifies to \(\frac{\partial w}{\partial z} = 0\) as \(w\) does not depend on \(z\).