Question

If \(f(x, y)=0,\) give the rule for finding \(d y / d x\) implicitly. If \(f(x, y, z)=0,\) give the rule for finding \(\partial z / \partial x\) and \(\partial z / \partial y\) implicitly.

Short answer

The rule for finding \(dy/dx\) implicitly when \(f(x, y)=0\) is \(- \frac{\partial f/ \partial x}{\partial f/ \partial y}\). Similarly, when \(f(x, y, z) = 0\), the rule for finding \(\partial z / \partial x\) and \(\partial z / \partial y\) implicitly is \(- \frac{\partial f/ \partial x + \frac{\partial f}{\partial y} \cdot \frac{dy}{dx}}{\partial f/\partial z}\) and \(- \frac{\partial f/ \partial y + \frac{\partial f}{\partial x} \cdot \frac{dx}{dy}}{\partial f/\partial z}\) respectively.

Step-by-step solution

Understand Implicit Differentiation

Implicit differentiation is the method used to differentiate equations that are not expressible as functions (not solved explicitly for a single variable). In this scenario, the task presents the functions in implicit form and asks for derivatives with respect to 'x' and 'y'. For both two and three variables, implicit differentiation involves applying the chain rule.

Rule for Differentiating Two Variable Equation

For an equation \(f(x, y) = 0\), applying implicit differentiation will give \(\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dx} = 0\). Setting this equation to 0 (as \(df = 0\)), and solving for \(dy/dx\), we get the rule \(\frac{dy}{dx} = - \frac{\partial f/ \partial x}{\partial f/ \partial y}\).

Rule for Differentiating Three Variable Equation

For an equation \(f(x, y, z) = 0\), applying implicit differentiation to find \(\partial z / \partial x\) gives \(\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dx} + \frac{\partial f}{\partial z} \cdot \frac{dz}{dx} = 0\). Using a similar procedure, we can solve this equation for \(\partial z / \partial x\), which gives \(\partial z / \partial x = - \frac{\partial f/ \partial x + \frac{\partial f}{\partial y} \cdot \frac{dy}{dx}}{\partial f/\partial z}\). A similar rule applies when finding \(\partial z / \partial y\).