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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.**

\[z_{x}-z_{y}=h(y)g'(x)-g(x)h'(y)\]