Second Order Derivatives

The derivative of a function is the fundamental concept in calculus. It represents the rate at which any given function is changing at a given point, or in simple words, it gives the slope of a function; this rate of change is also known as the first-order derivative of a function. Similarly, the Second Order Derivative represents the rate at which the first-order derivative is changing.

1.0Definition Of Second Order Derivative

To understand second-order derivatives, it is important to have a thorough understanding of the basics of First order derivatives:

• The first-order derivative is simply the rate at which a function is changing at any given point. It's like the slope of the tangent line on a graph. Think of it this way: if you were driving a car, the first derivative would tell you how fast you're going at any moment (your speed). It tells you how the function is changing in the most direct way. Mathematically, the first order derivative of a function, say f(x), is represented as: 

$f^{\prime }(x)$ or $\frac{d}{dx}[f(x)]$

• The second derivative takes it a step further, it is the derivative of the first order derivative, meaning it tells you how the rate of change is itself changing. If the first derivative is like your speed, the second derivative is like your acceleration, which lets you know if you're speeding up or slowing down. The second-order derivative can be represented as: 

f”(x) or $\frac{d}{dx}[f^{\prime }(x)]$ or $\frac{d^2}{d{x}^2}[f(x)]$

Second Order Derivative Formulas

There are no direct formulas for second-order derivatives; rather, the formulas can be derived from the existing formulas of first-order derivatives. For Example: 

1. Power Rule: if f(x) = xⁿ, then: 

f’(x) = nx^n-1, and f”(x) = n(n–1)x^n-2

2. Product Rule: If f(x) = g(x) . h(x), then: 

f'(x) = u'(x)v(x)+u(x)v'(x) and

f"(x)=g"(x)h(x)+2g'(x)h'(x)+g(x)h"(x)

3. Quotient Rule: If f(x) = u(x)/v(x), then: 

$f^{\prime }(x)=\frac{v(x)u^{\prime }(x)-u(x)v^{\prime }(x)}{v^2(x)}$

$f^{\prime \prime }(x)=\frac{v(x)[u^{\prime \prime }(x)v(x)-2u^{\prime }(x)]+u(x)[v^{\prime \prime }(x)v(x)-2v^{\prime }(x)]}{v^3(x)}$

Second-Order Derivatives of Parametric Functions: 

In calculus, parametric equations are a way of defining a curve in terms of two variables: one independent parameter, often denoted as (t), and two dependent variables x(t) & y(t).

To find the second-order derivative of a parametric function, use the derivatives of functions x(t) & y(t) with respect to "t" and further apply the chain rule to get the desired derivatives. 

For Example: Find the second-order derivative of the parametric equations: x = t² + 2t, y = 3t² - 5t 

Solution: 

Step 1: Find the first-order derivative of x and y with respect to t 

• $\frac{d}{dt}=\frac{d(t^2+2t)}{dt}=2t+2$
• $\frac{d}{dt}=\frac{d(3t^2-5t)}{dt}=9t+2$
• Hence, $\frac{d}{dx}=\frac{\frac{dy}{dx}}{\frac{dx}{dt}}=\frac{9t-2}{2t+2}$

Step 2: Next, Find the second-order derivative of dy/dx 

$\frac{d^{2}y}{d{x}^2}=\frac{\frac{d}{dt}(\frac{dy}{dx})}{\frac{dx}{dt}}=\frac{\frac{d}{dt}(\frac{9t-2}{2t+2})}{\frac{dx}{dt}}$

$\frac{d}{dt}(\frac{dy}{dx})=\frac{d}{dt}(\frac{9t-2}{2t+2})$

$\frac{d}{dt}(\frac{dy}{dx})=\frac{\frac{d}{dt}(9t-2)(2t+2)-\frac{d}{dt}(2t+2)(9t-2)}{(2t+2{)}^3}$

$\frac{d}{dt}(\frac{dy}{dx})=\frac{9(2t+2)-2(9t-2)}{(2t+2{)}^2}=\frac{18t+18-18t+4}{(2t+2{)}^2}=\frac{22}{(2t+2{)}^2}$

$\frac{d^{2}y}{d{x}^2}=\frac{22/(2t+2{)}^2}{(2t+2)}=\frac{22}{(2t+2{)}^3}$

Geometric Interpretation of Second-Order Derivative

Geometric interpretation of derivatives depends upon their order as the first-order derivative gives the slope of a tangent at a point while the second-order derivative represents the information about the concavity of the given function, such as: 

• If f"(x)>0, the graph is concave upwards as shown in the figure(the red and blue curves) like a “U”, the tangent’s slope increases. 
• If f"(x)<0, the graph is concave downwards, given the figure(the green and brown curves) like an “inverted U”, the tangent’s slope decreases.
• If f"(x)=0, the graph indicates an inflection point where the concavity changes.  

2.0Second Order Differential Equation

A differential equation is an equation which involves a function and its derivatives. A second-order differential equation is a special kind of differential equation which contains the second derivative of a function. In the case of a second-order differential equation, an unknown function y(x), together with its first and second derivatives, are involved in the equation. Mathematically, a second-order differential equation can be written as: 

y"(x)+p(x)y'(x)+q(x)y(x)=r(x)

Here, 

• y(x) is the unknown function we're trying to find (for example, the position of the car).
• y′(x) is the first derivative (the speed of the car).
• y′′(x) is the second derivative (the acceleration of the car).
• p(x),q(x), and r(x) are known functions (which depend on the problem you're solving).

3.0Second-Order Derivative Solved Examples

Problem 1: Given f(x) = 4x⁴ − 3x³ + 2x² − x, find the second-order derivative.

Solution: First order derivative of f(x): 

$f^{\prime }(x)=\frac{d}{dx}(4x^4-3x^3+2x^2-x)$

$=16x^3-9x^2+4x-1$

Compute the second derivative: 

$f"(x)=\frac{d}{dx}(16x^3-9x^2+4x-1)$

$=48x^2-18x+4$

Problem 2: Given f(x) = sin(x) . cos(x), find the second-order derivative. 

Solution: First order derivative of f(x):

$f^{\prime }(x)=\frac{d}{dx}(sinx+cosx)$

$=sinx\times \frac{d}{dx}(cosx)+\frac{d}{dx}(sinx)\times cosx$

$f^{\prime }(x)=sinx-sinx+cosx\times cosx$

$=cos2x-sin2x=cos2x$

Computing the second-order derivative: 

$f"(x)=\frac{d}{dx}(cos2x)$

$=-sin2x\times \frac{d}{dx}(2x)$

$=-2sin2x$

$=-2(2sinxcosx)=-4sinxcosx$

Problem 3: Find the second-order derivative of the function:

$f(x)=\frac{x^3}{e^x}$

Solution: First derivative by the quotient rule: 

$f^{\prime }(x)=\frac{d}{dx}(\frac{u(x)}{v(x)})=\frac{v(x)u^{\prime }(x)-u(x)v^{\prime }(x)}{v(x{)}^2}$

$f^{\prime }(x)=\frac{d}{dx}(\frac{x^3}{e^x})=\frac{e^x\times \frac{d}{dx}(x^3)-x^3\frac{d}{dx}(e^x)}{(e^x{)}^2}=\frac{e^x\times 3x^2-x^3\times (e^x)}{(e^x{)}^2}$

$f^{\prime }(x)=\frac{e^x(3x^2-x^3)}{(e^x{)}^2}=\frac{3x^2-x^3}{e^x}$

Compute the second-order derivative:

$f"(x)=\frac{d}{dx}(\frac{3x^2-x^3}{e^x})$

$f"(x)=(\frac{e^x\frac{d}{dx}(3x^2-x^3)-\frac{d}{dx}(e^x)(3x^2-x^3)}{(e^x{)}^2})$

$f"(x)=(\frac{e^x(6x-3x^2)-e^x(3x^2-x^3)}{(e^x{)}^2})$

$f"(x)=(\frac{e^x(6x-3x^2-3x^2+x^3)}{(e^{2x}))}=(\frac{e^x(6x-6x^2+x^3)}{(e^{2x})})$

$f"(x)=(\frac{6x-6x^2+x^3}{(e^x)})$