Higher Order Derivatives

The process of taking derivatives of derivatives refers to a higher-order derivative. There are many applications for higher-order derivatives, including sketching curves, solving motion problems, and many others. 

Such as

$\frac{d}{dx}$ [$\frac{d^{2}}{dx^{2}}$ [f(x)]] = $\frac{d^{3}}{dx^{3}}$ [f(x)]

$\frac{d}{dx}$ [$\frac{d^{3}}{dx^{3}}$ [f(x)]] = $\frac{d^{4}}{dx^{4}}$ [f(x)]

Example:

Evaluate the third derivative of 5x³ + 3x² + 7y

Solution

Step 1: Take the given function and apply the notation of differentiation to it and use laws.

$\frac{d}{dx}$ [f(x)] = $\frac{d}{dx}$ [5x³ + 3x² + 7y]

$\frac{d}{dx}$ [5x³ + 3x² + 7y] = $\frac{d}{dx}$ [5x³] + $\frac{d}{dx}$ [3x²] + $\frac{d}{dx}$ [7y]

Step 2: Now differentiate the above expression.

$\frac{d}{dx}$ [5x³ + 3x² + 7y] = [15x^3-1] + [6x^2-1] + [0]

$\frac{d}{dx}$ [5x³ + 3x² + 7y] = 15x² + 6x

Step 3: Now take the derivative of the first derivative.

$\frac{d}{dx}$ [$\frac{d}{dx}$ [5x³ + 3x² + 7y] = $\frac{d}{dx}$ [15x² + 6x]

$\frac{d}{dx}$ [$\frac{d}{dx}$ [5x³ + 3x² + 7y] = $\frac{d}{dx}$ [15x²] + $\frac{d}{dx}$ [6x]

$\frac{d^{2}}{dx^{2}}$ [5x³ + 3x² + 7y] = 30x^2-1 + 6x^1-1

$\frac{d^{2}}{dx^{2}}$ [5x³ + 3x² + 7y] = 30x¹ + 6x⁰

$\frac{d^{2}}{dx^{2}}$ [5x³ + 3x² + 7y] = 30x + 6

Step 4: Now take the derivative of the above second derivative of the given function.

$\frac{d}{dx}$ [$\frac{d^{2}}{dx^{2}}$ [5x³ + 3x² + 7y]] = $\frac{d}{dx}$ [30x + 6]

$\frac{d}{dx}$ [$\frac{d^{2}}{dx^{2}}$ [5x³ + 3x² + 7y]] = $\frac{d}{dx}$ [30x] + $\frac{d}{dx}$ [6]

$\frac{d^{3}}{dx^{3}}$ [5x³ + 3x² + 7y]] = [30x^1-1 + [0]

$\frac{d^{3}}{dx^{3}}$ [5x³ + 3x² + 7y]] = 30x⁰

$\frac{d^{3}}{dx^{3}}$ [5x³ + 3x² + 7y]] = 30(1) = 30

Hence, 30 is the third derivative (higher order derivative) of the given function.

There are a variety of derivatives that can be found in calculus. The most common derivatives are the derivative of a function with respect to one variable, the derivative of a function with respect to another variable, and the integral of a function. Each type of derivative has its own set of rules that must be followed in order to correctly take the derivatives.