Calculus/Higher Order Derivatives

> Differentiation

Introduction

The second derivative is the derivative of the derivative of a function. With the Newtonian notation, the derivative of the function $f(x)$ is denoted by $f^{\prime }(x)$, and its double (or "second") derivative is denoted by $f^{\prime \prime }(x)$. This is read as "f double prime of x," or "The second derivative of f(x)."

Because the derivative of function $f$ is defined as a function representing the slope of function $f$, the double derivative is the function representing the slope of the first derivative function. In Leibniz notation, this is written as:

$\frac{d}{dx}\left(\frac{df}{dx}\right)=\frac{d^{2}f}{d{x}^2}$.

The Newtonian notation is more common, yet the Leibniz notation is useful because of its precision.

Applications

The double derivative of a function has certain important uses. First, it allows one to calculate the concavity of a function at a point as follows:

• if $f^{\prime \prime }(x)>0$, the function $f$ is concave up (convex) at $x$.
• if $f^{\prime \prime }(x)<0$, the function $f$ is concave down at $x$.
• if $f^{\prime \prime }(x)=0$, the function $f$ may have a point of inflection at $x$. A point of inflection of $f$ is equivalent to a critical point of the first derivative of $f$.

The second derivative test is a test to see whether a critical point is a maximum or a minimum. If $f^{\prime }(x)=0$ at a certain point and, at the same point the second derivative is nonzero, then it is either a maximum or a minimum. If $f$ is concave up, then $f(x)$ is a minimum, and if it is concave down, then it is a maximum.

Examples

Example 1:

Find all points of inflection that occur on the function $y=x^3+4x.$

Solution:

• To find any points of inflection, one must know $\frac{d^{2}y}{d{x}^2}$ (aka. $y^{\prime \prime }$). In this example, one must find $\frac{dy}{dx}$ (aka. $y^{\prime }$) in order to find $\frac{d^{2}y}{d{x}^2}.$

$\frac{dy}{dx}=3x^2+4$

$\frac{d^{2}y}{d{x}^2}=6x$

• Set $\frac{d^{2}y}{d{x}^2}=0.$

$6x=0$

$x=0$

If there exists a point of inflection on the graph of $x^3+4x$ at $x=0$, then $\frac{d^{2}y}{d{x}^2}$ changes signs at $x=0$.

For $x<0,f^{″}(x)<0$

For $x>0,f^{″}(x)>0$

Therefore, $f(x)$ has a point of inflection at $x=0$.

Application to physics You can also take the derivative of a derivative of a derivative, or the third derivative of a function. This is useful in applications involving moving objects. For example, if $x(t)$ is a position function for a moving object, then

• $x(t)$ is its displacement function
• $v(t)=x^{\prime }(t)$ is its velocity function
• $a(t)=x^{\prime \prime }(t)$ is its acceleration function
• $j(t)=x^{\prime \prime \prime }(t)$ is its jerk function

Note:

The following section is being updated/moved. For more information, see Kinematics.

Example 2:

A particle moves along a path with a position that can be determined by the function $x(t)=4t^3+e^t$. 
Determine the acceleration when $t=3$.

Solution:

• Find $v(t)=\frac{dx}{dt}.$

$\frac{dx}{dt}=12t^2+e^t$

• Find $a(t)=\frac{dv}{dt}=\frac{d^{2}x}{d{x}^2}.$

$\frac{d^{2}x}{d{t}^2}=24t+e^t$

• Find $a(3)=\frac{d^{2}x}{d{t}^2}{|}_{t=3}$

$\frac{d^{2}x}{d{t}^2}{|}_{t=3}$	=	$24(3)+e^3$
	=	$72+e^3$
	=	$92.08553692...$

Third Derivatives and Beyond

Let $f(x)$ be a function in terms of x. The following are notations for higher derivatives. The far left is in Newtonian notation, and the middle and far right columns use Leibniz notation.

• First derviative: $f^{\prime }(x)=\frac{df}{dx}=\frac{d}{dx}[f(x)]$
• Second derivative: $f^{″}(x)=\frac{d^{2}f}{d{x}^2}=\frac{d^2}{d{x}^2}[f(x)]$
• Thrird derivative: $f^{‴}(x)=\frac{d^{3}f}{d{x}^3}=\frac{d^3}{d{x}^3}[f(x)]$
• Fourth derivative: $f^{(4)}(x)=\frac{d^{4}f}{d{x}^4}=\frac{d^4}{d{x}^4}[f(x)]$
• Nth derivative: $f^{(n)}(x)=\frac{d^{n}f}{d{x}^n}=\frac{d^n}{d{x}^n}[f(x)]$

Note: You should not write $f^n(x)$ to indicate the nth derivative, as this is easily confused with the quantity $f(x)$ all raised to the nth power.

One can derive a general expression for the nth derivative of $x^m$ where n and m are integers and n<m.

• $f(x)=x^m$
• $f^{\prime }(x)=(m)x^{m-1}$
• $f^{″}(x)=(m)(m-1)x^{m-2}$
• $f^{(3)}(x)=(m)(m-1)(m-2)x^{m-3}$

...

• $f^{(n)}(x)=\frac{m!}{(m-n)!}{x}^{m-n}$

Remember that $0!=1$, by definition.

Power, MacLaurin and Taylor series make use of this expression.

Example 1:

Find the third derivative of $f(x)=4x^5+6x^3+2x+1$ with respect to x.

Use the Power Rule to find the derivatives.

• $f^{\prime }(x)=20x^4+18x^2+2$
• $f^{″}(x)=80x^3+36x$
• $f^{‴}(x)=240x^2+36$

Example 2:

Find the third derivative of $g(x)=12e^x+\ln \left(x+2\right)+2x$ with respect to x.

Use the differentiation rules for exponential expressions, logarithmic expressions and polynomials.

• $g^{\prime }(x)=12e^x+\frac{1}{x+2}+2$
• $g^{″}(x)=12e^x-\frac{1}{(x+2{)}^2}$
• $g^{‴}(x)=12e^x+\frac{2}{(x+2{)}^3}$