# What's the difference between $\frac{\partial}{\partial x}$ and $\frac{d}{d x}$?

I wonder if there is a difference between: $$\frac{\partial f(x)}{\partial x}, \ \frac{\partial}{\partial x}f(x), \ \frac{d f(x)}{d x} \ \mathrm{and} \ \frac{d}{d x}f(x)$$ I saw all of them in a single demonstration and got quite confused.

• First of all, note that for understanding the difference between partial derivatives and total derivatives, you need to consider a function with more than one independent variable. I'm assuming a multi-variate function $$f$$.

$$\dfrac{\partial f}{\partial x}$$ and $$\dfrac{\partial}{\partial x}f$$ means same and they are the partial derivatives of the function $$f$$ with respect to $$x$$.

While, $$\dfrac{d f}{dx}$$ and $$\dfrac{d}{dx} f$$ means same and they are the derivatives of the function $$f$$ with respect to $$x$$.

The only difference between the derivative of a function and the partial derivative of a function is that, in case of partial derivatives, we have to keep all the independent variables as constants which is not to be considered in case of total derivative.

For example:

Let's say we have a function which is defined as $$f(x, y) = 2x + 3y$$

For this function, total derivative and partial derivatives w.r.t. $$x$$ are given by,

• $$\dfrac{df(x,y)}{dx} = \dfrac{d}{dx} (2x + 3y)= \dfrac{d}{dx} (2x) + \dfrac{d}{dx}(3y) = 2 + 3\dfrac{dy}{dx}$$
• $$\dfrac{\partial f}{\partial x} = \dfrac{\partial}{\partial x}(2x + 3y) =\dfrac{\partial}{\partial x}(2x) +\dfrac{\partial}{\partial x} (3y) = 2$$

In the first case, we have considered $$y$$ as a separate variable, and in the case of partial derivative, we have considered it as a constant.

In case the function has only one independent variable, then only: $$\dfrac{d f(x)}{dx} = \dfrac{d}{dx} f(x) = \dfrac{\partial f(x)}{\partial x} = \dfrac{\partial }{\partial x} f(x)$$

$$\dfrac{\partial f}{\partial x}$$ means partial derivative of $$f$$ wrt variable $$x$$ and it is used to indicate derivative of $$f$$ with respect to $$x$$ when $$f$$ has more than one independent variable, for example $$f(x,t)$$. If $$f$$ has only one independent variable as in $$f(x)$$, $$\dfrac{\partial f(x)}{\partial x}=\dfrac{df(x)}{dx}=\dfrac{\partial}{\partial x}f(x)=\dfrac{d}{dx}f(x)$$.