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The question aims to find the output based on a partial derivative using a given function. In mathematics, the output of one component of several variables is its output relative to one of those variables. At the same time, the other is kept constant (as opposed to the output of the total output, where all variables are allowed to vary). The partial derivative of a function for f(x,y,….) with respect to x is denoted by $f_{x}$, $f’_{x}$, $\partial_{x}$,$\dfrac{\partial f}{\partial x }$.It is also called the rate of change of a function with respect to $x$. It can be thought of as a change of function x-direction.
Expert Answer
Given $z=f(x)g(y)$
Step 1:When we find the partial derivative with respect to $x$, then $y$ is considered constant.
\[\dfrac{\partial}{\partial x}(h(x,y))=h_{x}(x,y)\]
\[\dfrac{\partial}{\partial x}(h(x,y))=z_{x}\]
When we find the partial derivative with respect to $y$, then $x$ is considered constant.
\[\dfrac{\partial}{\partial y}(h(x,y))=h_{x}(x,y)\]
\[\dfrac{\partial}{\partial y}(h(x,y))=z_{y}\]
Step 2: When we find the partial derivative of the given function with respect to $x$.
\[\dfrac{\partial z}{\partial x}=\dfrac{\partial }{\partial x}[f(x)g(y)]\]
\[z_{x}=g(y)f'(x)\]
When we find the partial derivative of the given function with respect to $y$.
\[\dfrac{\partial z}{\partial y}=\dfrac{\partial }{\partial y}[f(x)g(y)]\]
\[z_{y}=f(x)g'(y)\]
To find the value of $z_{x}+z_{y}$, plug values of partial derivatives.
\[z_{x}+z_{y}=g(y)f'(x)+f(x)g'(y)\]
Difference Between Derivative, Partial Derivative, and Gradient
Derivative
For the function has only one variable, derivatives are used.
example: $f (x) = 5x$, $f (z) = \sin (z) +3$
In the examples above $x$, and $z$ are variables. Since each function is a function of one variation, the output of the other can be used. Only one variable is used to differentiate the function.
\[f(x)=x^{5}\]
\[f'(x)=5x^{4}\]
Partial Derivative
The partial output is used when the function has two or more variables. The output of one component is considered relative to (w.r.t) one variable, while the other variables are considered the constant.
example: $f (x, y, z) = 2x + 3y + 4z$, where $x$, $y$, $z$ is a variable. The output of the partial one can be taken for each variable.
\[f(x,y,z)=2x+3y+4z\]
\[\partial f(x,y,z)=2\]
\[\dfrac{\partial f(x,y,z)}{\partial x}=2\]
\[\dfrac{\partial f(x,y,z)}{\partial y}=3\]
\[\dfrac{\partial f(x,y,z)}{\partial z}=4\]
The derivative is represented by $d$, while the derivative is represented as $\partial$.
Gradient
The gradient is a separate operator for functions with two or more variables. Gradient produces vector parts that come out as part of a function about its variance. Gradient combines everything that comes out of another part into a vector.
Numerical Result
The output of the $z_{x}+z_{y}$ is:
\[z_{x}+z_{y}=g(y)f'(x)+f(x)g'(y)\]
Example
First Partial Derivatives Given $z = g(x)h(y)$, find $z_{x}-z_{y}$.
Solution
Given $z=g(x)h(y)$
Step 1: When we calculate the partial derivative with respect to $x$, then $y$ is considered constant.
\[\dfrac{\partial}{\partial x}(g(x,y))=g_{x}(x,y)\]
\[\dfrac{\partial}{\partial x}(g(x,y))=z_{x}\]
When we find the partial derivative with respect to $y$, then $x$ is considered constant.
\[\dfrac{\partial}{\partial y}(g(x,y))=g_{x}(x,y)\]
\[\dfrac{\partial}{\partial y}(g(x,y))=z_{y}\]
Step 2: When we find the partial derivative of the given function with respect to $x$.
\[\dfrac{\partial z}{\partial x}=\dfrac{\partial }{\partial x}[g(x)h(y)]\]
\[z_{x}=h(y)g'(x)\]
When we find the partial derivative of the given function with respect to $y$.
\[\dfrac{\partial z}{\partial y}=\dfrac{\partial}{\partial y}[g(x)h(y)]\]
\[z_{y}=g(x)h'(y)\]
To find the value of $z_{x}-z_{y}$, plug values of partial derivatives.
\[z_{x}-z_{y}=h(y)g'(x)-g(x)h'(y)\]