Is there a difference operation $\frac{d}{dx}$ and $'$?

Question

I have a question I've been wondering about for a long time.

Is operation $«\color{red}{\frac{d}{dx}}»$ mathematically equal to operation $«\color{red}{'}»$ ?

Is there any difference between them?

Answer 1

There is no mathematical difference. They are different notations for the same thing. (The first was due to Newton, the second to Leibniz.)

Answer 2

Using prime ' usually indicates that we take the total derivative w.r.t. to all variables, whereas $\frac{d}{dx}$ indicates that we take the total derivative with respect to the variable $x$. Note that this is different from the partial derivative $\frac{\partial }{\partial x}$.

As an example, consider $f(x, y, z)=x^2+y^2+3z$ where $y=\sin(x)$. Then

• $f' = (2x, 2y, 3) $ (the Jacobian matrix whose entries are the partial derivatives)
• $\frac{d}{dx}f = 2x + 2y\cos(x) $
• $\frac{\partial}{\partial x}f = 2x$

Answer 3

There is no difference between the notations - they mean exactly the same thing. However, at different times you will find one more useful than the other. For example, when doing u-substitution with integrals, the $\frac{d}{dx}$ is helpful. The same thing is true when using the chain rule - it is often easier to keep track of what is happening with $\frac{d}{dx}$. But, writing " ' " is definitely quicker, more efficient, and sometimes neater.