Is this the complete problem statement? Generally speaking you don't find "∂z/∂x" of something, and more than you find "dy/dx" of something. You can, however, find the derivative with respect to, say, x of a function (denoted d/dx(f)) or the partial of some function with respect to, say z (denoted ∂/∂x(f).Calpalned said:Homework Statement
Find (∂z/∂x) of 6xyz
Calpalned said:Homework Equations
N/a
The Attempt at a Solution
The correct answer is 6xy(∂z/∂x) but I would like proof of it. I got something different when I tried taking the partial derivative.
6xyz = 6x(yz) = Multiplication rule for derivatives
6(∂x/∂x) + y(∂z/∂x)
What did I do wrong? Thanks
The derivative above is from my textbook's solutions guide.Calpalned said:The full question is find (∂z/∂x) of x3 + y3 + z3 + 6xyz = 1
My textbook says the partial derivative is 3x2 + 3z2(∂z/∂x) + 6xy(∂z/∂x) = 0
I don't get how to take the derivative of the red part.
Corrected to: find (∂z/∂x)Calpalned said:The full question is find (∂z/∂x) of x3 + y3 + z3 + 6xyz = 1
My textbook says the partial derivative is 3x2 + 3z2(∂z/∂x) + 6xy(∂z/∂x) = 0
I don't get how to take the derivative of the red part.
An implicit partial derivative is a mathematical concept that describes the rate of change of a function with respect to one of its independent variables, while holding all other independent variables constant. It is used when a function is expressed implicitly, rather than explicitly, as a function of its independent variables.
An explicit partial derivative is calculated by directly differentiating a function with respect to one of its independent variables. In contrast, an implicit partial derivative involves differentiating an equation that relates multiple variables, and therefore requires the use of implicit differentiation.
Implicit partial derivatives are important because they allow us to analyze the behavior of functions that are expressed implicitly, which is often the case in real-world applications. They also help us to understand the relationships between different variables in a function.
To calculate an implicit partial derivative, you must use the chain rule to differentiate each term in the equation with respect to the variable of interest. The resulting expression will include both the partial derivative of the function and the derivative of the variable with respect to which it is being differentiated.
Implicit partial derivatives are commonly used in fields such as physics, engineering, economics, and statistics, where functions are often expressed implicitly. They are also important in optimization and modeling processes, as they allow us to analyze the behavior of complex systems with multiple variables.