Use implicit differentiation to find
dz/dy for yz = ln (x+z)
I am unsure of how to deal with the right side specifically.
Consider z as a function of y, and differentiate both sides with respect to y.
Is x supposed to be a constant on the right? Or did you mean to type "y" there? If x is another variable, z is not a ntion of y alone, and you would have to deal with partial derivatives.
y dz/dy + z dy/dy = dz/dy /(x+z)
z = (dz/dy)(1/(x+z) - y)
z = (dz/dy) ( 1 -zx - xy)/(x+z)
dz/dy = (z^2 + xz)/(1 - xz - xy)
I assumed that partial is intended and x is constant for dz/dy
To find dz/dy for the given equation yz = ln(x+z) using implicit differentiation, we need to differentiate both sides of the equation with respect to y while treating x as a constant.
Let's start by differentiating the left side of the equation: yz.
To differentiate yz, we consider y as a function of y itself. Thus, we get the derivative as follows:
d(yz)/dy = y * dz/dy + z * dy/dy (using the product rule)
The derivative dy/dy is simply 1, so the equation becomes:
yz = y * dz/dy + z * 1
Next, let's differentiate the right side of the equation: ln(x+z).
To do this, we consider ln(x+z) as a composite function of y, since both x and z could depend on y. The derivative of ln(x+z) with respect to y can be found using the chain rule:
d(ln(x+z))/dy = (1/(x+z)) * d(x+z)/dy
To find d(x+z)/dy, we differentiate x+z with respect to y, treating x as a constant and z as a function of y. This gives us:
d(x+z)/dy = 0 + dz/dy
Now, we can substitute this result back into our equation:
d(ln(x+z))/dy = (1/(x+z)) * (dz/dy)
So, our original equation becomes:
yz = y * dz/dy + z
Now we can solve this equation for dz/dy:
y * dz/dy = yz - z
Finally, we can find dz/dy by dividing both sides of the equation by y:
dz/dy = (yz - z) / y
Therefore, the expression for dz/dy for the given equation yz = ln(x+z) is (yz - z) / y.