What is the difference among dp/dx , ∂p/∂x , ∆x with example. Engineering Mathematics

What is the difference among dp/dx , ∂p/∂x , ∆x with example. Engineering Mathematics 

1. $\frac{��}{��}$: This represents the derivative of a function $�$ with respect to $�$. It denotes the rate of change of $�$ with respect to $�$, which is essentially the slope of the tangent line to the curve of $�$ with respect to changes in $�$. This notation is typically used when dealing with continuous functions.

   Example: Let's say $�(�)=3{�}^2+2�+1$. Then, $\frac{��}{��}$ would be the derivative of $�(�)$ with respect to $�$, which is $\frac{��}{��}=6�+2$.

1. $\frac{\mathrm{\partial }�}{\mathrm{\partial }�}$: This represents the partial derivative of a function $�$ with respect to $�$. It's used when dealing with multivariable functions, where $�$ might depend on several variables, but we're only interested in how it changes concerning $�$ while holding other variables constant.

   Example: Consider $�(�,�)={�}^2+2��+{�}^2$. The partial derivative $\frac{\mathrm{\partial }�}{\mathrm{\partial }�}$ would be $2�+2�$, indicating how $�$ changes concerning $�$ while $�$ remains constant.

1. $\mathrm{\Delta }�$: This represents the change in $�$, often denoted as "delta x." It's the difference between two values of $�$, indicating the displacement along the x-axis.

   Example: Suppose we have a function $�(�)={�}^2$ and we want to find the change in $�$ corresponding to a change in $�$ from $�=2$ to $�=4$. Then, $\mathrm{\Delta }�=4-2=2$.

In summary, while $\frac{��}{��}$ and $\frac{\mathrm{\partial }�}{\mathrm{\partial }�}$ denote derivatives with respect to $�$, the former is for single-variable functions, and the latter is for multivariable functions. $\mathrm{\Delta }�$, on the other hand, simply represents the change in the value of $�$, regardless of whether we're dealing with single or multiple variables.