Second Order Derivatives

Second derivatives are obtained by taking the derivative of a given quantity with respect to another variable. In order words, the derivative of the first derivative is referred to as the second order derivative. Such as:

$\frac{d}{dx}$ [$\frac{d}{dx}$ (f(x))] = d²/dx² [f(x)]

For example

Evaluate the second derivative of 3x⁴ + cos(x)

Solution

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

$\frac{d}{dx}$ [f(x)] = $\frac{d}{dx}$ [3x⁴ + cos(x)]

$\frac{d}{dx}$ [3x⁴ + cos(x)] = $\frac{d}{dx}$ [3x⁴] + $\frac{d}{dx}$ [cos(x)]

Step 2: Now differentiate the above expression.

$\frac{d}{dx}$ [3x⁴ + cos(x)] = [12x^4-1] + [-sin(x)]

$\frac{d}{dx}$ [3x⁴ + cos(x)] = 12x³ – sin(x)

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

$\frac{d}{dx}$ [$\frac{d}{dx}$ [3x⁴ + cos(x)]] = $\frac{d}{dx}$ [12x³ – sin(x)]

d²/dx² [3x⁴ + cos(x)] = $\frac{d}{dx}$ [12x³] – $\frac{d}{dx}$ [sin(x)]

d²/dx² [3x⁴ + cos(x)] = 36x^3-1 – [cos(x)]

d²/dx² [3x⁴ + cos(x)] = 36x² – cos(x)

Hence, 36x² – cos(x) is the second derivative of the given function.