Derivative Notation

Lagrange Notation: $f'(x)$

Prime notation is the most common in calculus.

• $f'(x)$: First derivative
• $f''(x)$: Second derivative (derivative of the derivative)
• $f'''(x)$: Third derivative
• $f^{(n)}(x)$: nth derivative

Example: If $f(x) = x^3$, then $f'(x) = 3x^2$

Leibniz Notation: $\frac{dy}{dx}$

This looks like a fraction, and sometimes we can treat it like one!

Parts:

• $dy$: "Infinitesimal change in y"
• $dx$: "Infinitesimal change in x"
• $\frac{dy}{dx}$: "Rate of change of y with respect to x"

Example: If $y = x^2$, then $\frac{dy}{dx} = 2x$

At a Specific Point

To indicate the derivative at a particular value:

Lagrange: $f'(3) \text{ or } f'(a)$

Leibniz: $\left.\frac{dy}{dx}\right|_{x=3} \text{ or } \frac{dy}{dx}\bigg|_{x=a}$

The vertical bar means "evaluated at".

Higher-Order Derivatives

Second derivative (derivative of the derivative):

Third derivative:

• $f'''(x)$ or $f^{(3)}(x)$
• $\frac{d^3y}{dx^3}$

nth derivative:

• $f^{(n)}(x)$
• $\frac{d^ny}{dx^n}$

Why Different Notations?

Use $f'(x)$ when:

• Working with function notation
• You want clean, simple expressions

Use $\frac{dy}{dx}$ when:

• Emphasizing the relationship between variables
• Using the chain rule
• Doing implicit differentiation
• Separating variables in differential equations

Use $\dot{y}$ when:

• Dealing with time derivatives in physics
• $t$ is the independent variable

Example Comparison

For $y = x^3 + 2x$:

Prime notation: $f'(x) = 3x^2 + 2$

Leibniz notation: $\frac{dy}{dx} = 3x^2 + 2$

At $x = 1$: $f'(1) = 3(1)^2 + 2 = 5$ $\left.\frac{dy}{dx}\right|_{x=1} = 5$

They all give the same answer!

The "d" Operator

When we write $\frac{d}{dx}$, we can think of it as an operator:

$\frac{d}{dx}[x^2] = 2x$

Read as: "Take the derivative with respect to x of $x^2$"

Partial Derivatives (Preview)

For functions of multiple variables (later in calculus):

$\frac{\partial f}{\partial x}$

The $\partial$ symbol indicates a partial derivative (holding other variables constant).

Practice Tips

1. Be consistent within a problem
2. Read the question - match the notation used
3. $f'(x)$ is faster for simple work
4. $\frac{dy}{dx}$ is clearer for chain rule and related rates
5. Context matters - physics vs. pure math