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Use implicit differentiation to find
(del z)/(del x) and
(del z)/(del y).
{:[yz=8ln(x+z)],[(del z)/(del x)=◻],[(del z)/(del y)=◻]:}](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/solver-b2c/aae71be4-e15a-11ee-94f9-8ecbbdcafd9b.png)
. Points]DETAILSPREVIOUS ANSWERSSCALCEUse implicit differentiation to find and .
Full solution
Q. . Points]DETAILSPREVIOUS ANSWERSSCALCEUse implicit differentiation to find and .
- Differentiate with respect to x: We are given the equation . To find , we need to differentiate both sides of the equation with respect to x, treating y as a constant.Differentiating the left side with respect to x gives us , since y is treated as a constant.Differentiating the right side with respect to x gives us , using the chain rule for the natural logarithm function and the sum inside it.Setting up the differentiation: .
- Solve for (\frac{\partial z}{\partial x}): Now we solve for \((\frac{\partial z}{\partial x})\. We have \(y\cdot(\frac{\partial z}{\partial x}) = \frac{\(8\)}{x+z} + \frac{\(8\)}{x+z}\cdot(\frac{\partial z}{\partial x})\. Rearrange the terms to isolate \((\frac{\partial z}{\partial x}): \(y\cdot(\frac{\partial z}{\partial x}) - \frac{\(8\)}{x+z}\cdot(\frac{\partial z}{\partial x}) = \frac{\(8\)}{x+z}\. Factor out \((\frac{\partial z}{\partial x}): \((\frac{\partial z}{\partial x}) \cdot (y - \frac{\(8\)}{x+z}) = \frac{\(8\)}{x+z}\. Divide both sides by \((y - \frac{\(8\)}{x+z}) to solve for \((\frac{\partial z}{\partial x}): \((\frac{\partial z}{\partial x}) = \frac{\(8\)}{x+z} / (y - \frac{\(8\)}{x+z})\. Simplify the expression: \((\frac{\partial z}{\partial x}) = \frac{\(8\)}{(x+z)(y - \frac{\(8\)}{x+z})}\.
- Differentiate with respect to \(y: Next, we find by differentiating both sides of the original equation with respect to , treating as a constant.Differentiating the left side with respect to gives us , using the product rule.Differentiating the right side with respect to gives us , since there are no terms in the natural logarithm function.Setting up the differentiation: .
- Solve for (\frac{\partial z}{\partial y}): Now we solve for \$(\frac{\partial z}{\partial y})\. We have \$z + y\cdot(\frac{\partial z}{\partial y}) = 0\. Subtract from both sides: y\cdot(\frac{\partial z}{\partial y}) = -z\. Divide both sides by to solve for (\frac{\partial z}{\partial y}): \$\frac{\partial z}{\partial y} = -\frac{z}{y}\.
