Higher Order Derivatives and Leibnitz Theorem

Introduction to Higher Order Derivatives

We have already learned how to find the first derivative of a function, which represents the instantaneous rate of change of the function. Higher-order derivatives are simply the derivatives of a function taken more than once. The second derivative is the derivative of the first derivative, the third derivative is the derivative of the second derivative, and so on. These higher-order derivatives provide crucial information about the behavior of a function, such as its concavity, points of inflection, and local extrema.

Second Derivative

Let $y = f(x)$ be a differentiable function. The first derivative of $y$ with respect to $x$ is denoted by $\frac{dy}{dx}$ or $f'(x)$ or $y'$. If the first derivative, $\frac{dy}{dx}$, is itself differentiable, then its derivative is called the second derivative of $y$ with respect to $x$. The second derivative is denoted in various ways:

• $\frac{d^2y}{dx^2}$
• $f''(x)$
• $y''$

In essence, $\frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right)$.

Example of Second Derivative Calculation

Let's find the second derivative of the function $f(x) = x^3 + 2x^2 - 5x + 1$. First, we find the first derivative: $f'(x) = \frac{d}{dx}(x^3 + 2x^2 - 5x + 1) = 3x^2 + 4x - 5$. Now, we differentiate $f'(x)$ to find the second derivative: $f''(x) = \frac{d}{dx}(3x^2 + 4x - 5) = 6x + 4$. So, the second derivative of $f(x) = x^3 + 2x^2 - 5x + 1$ is $f''(x) = 6x + 4$.

Third Derivative

The third derivative is the derivative of the second derivative. It is denoted by:

• $\frac{d^3y}{dx^3}$
• $f'''(x)$
• $y'''$

$\frac{d^3y}{dx^3} = \frac{d}{dx} \left( \frac{d^2y}{dx^2} \right)$.

Example of Third Derivative Calculation

Using the function from the previous example, $f(x) = x^3 + 2x^2 - 5x + 1$, we found $f''(x) = 6x + 4$. Now, we find the third derivative by differentiating $f''(x)$: $f'''(x) = \frac{d}{dx}(6x + 4) = 6$. The third derivative is a constant, 6.

nth Order Derivative

This process can be continued to find the fourth, fifth, and so on, up to the nth derivative. The nth derivative of $y = f(x)$ is denoted by:

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

$\frac{d^ny}{dx^n} = \frac{d}{dx} \left( \frac{d^{(n-1)}y}{dx^{(n-1)}} \right)$.

Derivatives of Simple Functions

Let's look at the higher-order derivatives of some common functions:

1. If $f(x) = x^m$, then: $f'(x) = mx^{m-1}$ $f''(x) = m(m-1)x^{m-2}$ $f'''(x) = m(m-1)(m-2)x^{m-3}$ ... $f^{(n)}(x) = m(m-1)(m-2)...(m-n+1)x^{m-n}$ This can be written using the falling factorial notation as $f^{(n)}(x) = P(m, n) x^{m-n}$, where $P(m, n) = \frac{m!}{(m-n)!}$. If $m$ is a non-negative integer, then for $n > m$, $f^{(n)}(x) = 0$.
2. If $f(x) = e^{ax}$, then: $f'(x) = ae^{ax}$ $f''(x) = a^2e^{ax}$ $f'''(x) = a^3e^{ax}$ ... $f^{(n)}(x) = a^n e^{ax}$
3. If $f(x) = \sin(ax + b)$, then: $f'(x) = a\cos(ax + b)$ $f''(x) = -a^2\sin(ax + b)$ $f'''(x) = -a^3\cos(ax + b)$ $f^{(4)}(x) = a^4\sin(ax + b)$ The pattern repeats every four derivatives. We can express the nth derivative as: $f^{(n)}(x) = a^n \sin\left(ax + b + \frac{n\pi}{2}\right)$
4. If $f(x) = \cos(ax + b)$, then: $f'(x) = -a\sin(ax + b)$ $f''(x) = -a^2\cos(ax + b)$ $f'''(x) = a^3\sin(ax + b)$ $f^{(4)}(x) = a^4\cos(ax + b)$ Similarly, the nth derivative is: $f^{(n)}(x) = a^n \cos\left(ax + b + \frac{n\pi}{2}\right)$
5. If $f(x) = \ln(x)$, then: $f'(x) = \frac{1}{x} = x^{-1}$ $f''(x) = -x^{-2} = -\frac{1}{x^2}$ $f'''(x) = 2x^{-3} = \frac{2}{x^3}$ $f^{(4)}(x) = -6x^{-4} = -\frac{6}{x^4}$ The general form for the nth derivative of $\ln(x)$ is: $f^{(n)}(x) = (-1)^{n-1} (n-1)! x^{-n} = \frac{(-1)^{n-1}(n-1)!}{x^n}$ for $n \ge 1$.

Leibnitz Theorem for Higher Order Derivatives of a Product

When we need to find the nth derivative of a product of two functions, say $y = uv$, where $u$ and $v$ are functions of $x$, we can use the Leibnitz Theorem. This theorem is a generalization of the product rule for differentiation.

The theorem states that if $y = uv$, then the nth derivative of $y$ is given by: $\frac{d^ny}{dx^n} = \sum_{r=0}^{n} \binom{n}{r} \frac{d^{n-r}u}{dx^{n-r}} \frac{d^rv}{dx^r}$

Let's expand this summation to understand it better: $\frac{d^ny}{dx^n} = \binom{n}{0} u^{(n)} v^{(0)} + \binom{n}{1} u^{(n-1)} v^{(1)} + \binom{n}{2} u^{(n-2)} v^{(2)} + ... + \binom{n}{n-1} u^{(1)} v^{(n-1)} + \binom{n}{n} u^{(0)} v^{(n)}$

Where:

• $u^{(k)}$ denotes the kth derivative of $u$ with respect to $x$.
• $v^{(k)}$ denotes the kth derivative of $v$ with respect to $x$.
• $v^{(0)}$ is simply $v$ (the 0th derivative).
• $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ are the binomial coefficients.

The binomial coefficients $\binom{n}{r}$ follow the pattern of Pascal's triangle.

Example of Leibnitz Theorem Application

Let's find the third derivative of $y = x^2 e^x$. Here, we can set $u = e^x$ and $v = x^2$. We need to find the derivatives of $u$ and $v$ up to the third order.

Derivatives of $u = e^x$: $u^{(0)} = e^x$ $u^{(1)} = e^x$ $u^{(2)} = e^x$ $u^{(3)} = e^x$ In general, $u^{(n)} = e^x$.

Derivatives of $v = x^2$: $v^{(0)} = x^2$ $v^{(1)} = 2x$ $v^{(2)} = 2$ $v^{(3)} = 0$ For $r \ge 3$, $v^{(r)} = 0$.

Now, apply Leibnitz Theorem for $n=3$: $\frac{d^3y}{dx^3} = \sum_{r=0}^{3} \binom{3}{r} u^{(3-r)} v^{(r)}$ $\frac{d^3y}{dx^3} = \binom{3}{0} u^{(3)} v^{(0)} + \binom{3}{1} u^{(2)} v^{(1)} + \binom{3}{2} u^{(1)} v^{(2)} + \binom{3}{3} u^{(0)} v^{(3)}$

Calculate the binomial coefficients: $\binom{3}{0} = 1$ $\binom{3}{1} = 3$ $\binom{3}{2} = 3$ $\binom{3}{3} = 1$

Substitute the derivatives and coefficients: $\frac{d^3y}{dx^3} = (1)(e^x)(x^2) + (3)(e^x)(2x) + (3)(e^x)(2) + (1)(e^x)(0)$ $\frac{d^3y}{dx^3} = e^x x^2 + 6x e^x + 6 e^x + 0$ $\frac{d^3y}{dx^3} = e^x (x^2 + 6x + 6)$

Another Example: Finding the nth derivative

Find the nth derivative of $y = x \sin x$. Let $u = \sin x$ and $v = x$.

Derivatives of $u = \sin x$: $u^{(n)} = \sin\left(x + \frac{n\pi}{2}\right)$

Derivatives of $v = x$: $v^{(0)} = x$ $v^{(1)} = 1$ $v^{(2)} = 0$ For $r \ge 2$, $v^{(r)} = 0$.

Apply Leibnitz Theorem for the nth derivative: $\frac{d^ny}{dx^n} = \sum_{r=0}^{n} \binom{n}{r} u^{(n-r)} v^{(r)}$ Since $v^{(r)}$ is zero for $r \ge 2$, the sum only goes up to $r=1$: $\frac{d^ny}{dx^n} = \binom{n}{0} u^{(n)} v^{(0)} + \binom{n}{1} u^{(n-1)} v^{(1)}$ $\frac{d^ny}{dx^n} = (1) \sin\left(x + \frac{n\pi}{2}\right) (x) + (n) \sin\left(x + \frac{(n-1)\pi}{2}\right) (1)$

We know that $\sin\left(\theta + \frac{\pi}{2}\right) = \cos(\theta)$ and $\sin\left(\theta - \frac{\pi}{2}\right) = -\cos(\theta)$. Also, $\sin\left(\alpha + (n-1)\frac{\pi}{2}\right) = \sin\left(\alpha + n\frac{\pi}{2} - \frac{\pi}{2}\right)$. Let $\theta = x + \frac{n\pi}{2}$. Then $\sin\left(x + \frac{(n-1)\pi}{2}\right) = \sin\left(\theta - \frac{\pi}{2}\right) = -\cos\left(\theta\right) = -\cos\left(x + \frac{n\pi}{2}\right)$.

So, the nth derivative becomes: $\frac{d^ny}{dx^n} = x \sin\left(x + \frac{n\pi}{2}\right) - n \cos\left(x + \frac{n\pi}{2}\right)$

Applications of Higher Order Derivatives

Higher-order derivatives are fundamental in many areas of mathematics, physics, and engineering.

• Taylor and Maclaurin Series: These series expansions of functions rely heavily on higher-order derivatives. They allow us to approximate complex functions with simpler polynomials, which is essential for numerical methods and theoretical analysis. The Taylor series expansion of a function $f(x)$ around a point $a$ is given by: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$ If $a=0$, it is the Maclaurin series.
• Curve Sketching: The second derivative ($f''(x)$) tells us about the concavity of a function.
  • If $f''(x) > 0$ on an interval, the function is concave up (like a smile).
  • If $f''(x) < 0$ on an interval, the function is concave down (like a frown).
  Points where the concavity changes are called inflection points. These occur where $f''(x) = 0$ or is undefined, and the sign of $f''(x)$ changes around that point.
• Optimization: The second derivative test is used to classify critical points (where $f'(x) = 0$) as local maxima or minima.
  • If $f'(c) = 0$ and $f''(c) < 0$, then $f$ has a local maximum at $c$.
  • If $f'(c) = 0$ and $f''(c) > 0$, then $f$ has a local minimum at $c$.
  • If $f'(c) = 0$ and $f''(c) = 0$, the test is inconclusive.
• Physics: In mechanics, the position of an object is often denoted by $s(t)$. The first derivative, $v(t) = s'(t)$, is the velocity. The second derivative, $a(t) = v'(t) = s''(t)$, is the acceleration. Higher derivatives can describe jerk, snap, and other concepts related to the rate of change of motion.
• Differential Equations: Many physical laws are expressed as differential equations, which involve derivatives of unknown functions. Higher-order derivatives are crucial for formulating and solving these equations, which model phenomena like vibration, heat transfer, and wave propagation.

Summary Table of Higher Derivatives

Here's a quick reference for the nth derivatives of common functions: