Why are there different notations for derivatives, like f'(x) and dy/dx?

The existence of multiple notations for derivatives is a direct result of calculus being independently invented by two different mathematicians: Isaac Newton and Gottfried Wilhelm Leibniz. Both of their notations, along with a third from Lagrange, have survived because each offers unique advantages in different contexts.

Think of them not as competing systems, but as different tools designed for specific tasks.

• Leibniz's Notation: dy/dx This is perhaps the most descriptive notation. The d is meant to signify a "delta" or a tiny, infinitesimal change. Therefore, dy/dx can be read as "the infinitesimal change in y with respect to the infinitesimal change in x."
• Strengths: This notation is excellent for explicitly stating the variable you are differentiating with respect to (in this case, x). This becomes crucial in multivariable calculus and when solving differential equations. It also makes the chain rule wonderfully intuitive.
• Lagrange's Notation: f'(x) Developed by Joseph-Louis Lagrange, this is a more compact and function-focused notation. The prime symbol (') indicates that the function f has been differentiated once. You read it as "f prime of x." The second derivative is simply f''(x).
• Strengths: This notation is clean and efficient, especially when you need to plug a specific number into a derivative. Writing f'(3) is much neater and clearer than the Leibniz equivalent for finding the derivative's value at x=3.
• Newton's Notation: ? Newton, who was primarily a physicist, was focused on how things change over time. He used a dot over a variable to represent its derivative with respect to time.
• Strengths: While you won't see it as often in a general math class, "dot notation" is still widely used in physics, mechanics, and engineering as a quick shorthand for time derivatives.

Key Takeaway: You are not meant to choose one notation over the others. A skilled student of calculus learns to be fluent in both Leibniz (dy/dx) and Lagrange (f'(x)) notations and understands when to use each one for maximum clarity and efficiency.