The way you write taking the derivative of $f(x)$

$$\frac{d}{dx}f(x)$$

It's important to know $\frac{d}{dx}$ is an OPERATOR that is applied to $f(x)$

The $dx$ in the denominator means we are differentiating in respect to the variable $x$

For Example if we had a function for distance in respect to time, $s(t)$, we would take the derivative by:

$$\frac{d}{dt} s(t)$$

Here are the different ways to write the derivative for $f(x)$

This is the final answer, not the operation

Lagrange's Notation:

$$f'(x) \text{ or } y'$$

Leibniz's Notation:

$$\frac{dy}{dx} \text{ or } \frac{df(x)}{dx}$$

Euler's Notation:

$$D_{x}f(x)$$

Newtons's Notation:

$$\dot{y}$$

Euler's and Newton's notation is rarely ever used

Lagrange's Notation is the most common for single variable calculus (like calculus 1)

Leibniz's Notation is more common in physics and multi-variable calculus

Leibniz notation also can make remembering calculus rules easier (by treating it like a fraction)

We will be using Lagrange's notation most in these notes

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Higher Order Derivatives

Since the derivative of a function outputs another function, we can also take the derivative of this function

Taking the derivative of the derivative is called the 2nd derivative

Taking the derivative of a function n times is called the nth derivative

There are many reasons why you would need to take the derivative multiple times

For Example, if $s(t)$ is distance, then the derivative of $s(t)$ is velocity, and the derivative of the derivative of $s(t)$ is acceleration

Here is the OPERATOR for taking the nth derivative of $f(x)$

$$\frac{d^n}{{dx}^n}f(x)$$

Here are the different ways to write the nth derivative for $f(x)$

Lagrange's Notation:

$$f''(x) \text{ or } y''$$

$$f'''(x) \text{ or } y'''$$

$$f^{(4)}(x) \text{ or } y^{(4)}$$

You keep adding tick marks (') until you get 4 or higher where you then just write the nth in parenthese

Leibniz's Notation:

$$\frac{d^{n}y}{{dx}^n} \text{ or } \frac{d^{n}f(x)}{{dx}^n}$$

Euler's Notation:

$$D_{x}^{n}f(x)$$

Newtons's Notation:

$$\ddot{y}$$

$$\dddot{y}$$

You keep adding dots (.) on top until you get 4 or higher where you then just write the nth on top

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Evaluating At A Point

Since the derivative returns a new function that can be evaluated at a point, it is useful to have a way to denote derivative at a point

For a point $x = c$

Lagrange notation:

$$f'(c)$$

Leibniz notation:

$$\frac{dy}{dx} \Biggr|_{x=c}$$

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Practice Problems

• If the derivative of $f(x) = 2x^3 + 4x^2 - x + 1$ is $6x^2 + 8x - 1$, what are the ways to write this?

  $$f'(x) = 6x^2 + 8x - 1$$

  $$\frac{dy}{dx} = 6x^2 + 8x - 1$$

  $$D_{x}f(x) = 6x^2 + 8x - 1$$

  $$\dot{y} = 6x^2 + 8x - 1$$

  If the derivative of $f(x) = x^9 + 4x^4 + x - 1$ is $9x^8 + 16x^3 + 1$, write this in Lagrange and Leibniz notation

  $$f'(x) = 9x^8 + 16x^3 + 1$$

  $$\frac{dy}{dx} = 9x^8 + 16x^3 + 1$$

  Rewrite $\frac{d}{dx}x^2 = 2x$ in Lagrange and Leibniz notation

  $$f'(x) = 2x$$

  $$\frac{dy}{dx} = 2x$$

  If the 2nd derivative of $f(x) = 3x^3 + 2x^2 + x$ is $18x + 4$, write this in Lagrange and Leibniz notation

  $$f''(x) = 18x + 4$$

  $$\frac{d^2y}{dx^2} = 18x + 4$$

  Rewrite $f'(x) = 4x^2$ in Leibniz notation and using the derivative operator

  $$\frac{d}{dx}f(x) = 4x^2$$

  $$\frac{dy}{dx} = 4x^2$$

  Rewrite $f^{4}(x) = 2$ in Leibniz notation and using the derivative operator

  $$\frac{d^4}{dx^4}f(x) = 2$$

  $$\frac{d^4y}{dx^4} = 2$$

  Rewrite $\frac{d^3y}{dx^3} = x + 1$ in Lagrange notation and using the derivative operator

  $$f'''(x) = x + 1$$

  $$\frac{d^3}{dx^3}f(x) = x+1$$