Higher order Derivatives

In the following formulas, we show the higher order derivatives before diving into practical applications.

If we define functions Like $f(x)$ , $v(x)$ , $y(x)$ and $u(x)$ with $x$ being the variable and $n$ a natural number:

$\frac{d}{dx}(u^{v})=vu^{v-1}\cdot \frac{du}{dx}+u^{v}\ln u \cdot\frac{dv}{dx}$

The chain rule:

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

This can be used to generate many formulas seen in our table.

Example:

$y=(5x-8)^{4}$

Let $u=5x-8$

We know that:

$(u^{4})'=4u^{3}u'$

$\frac{du}{dx}=\frac{d}{dx}(5x-8)=5$

Finally:

$\frac{d}{dx}((5x-8)^{4})=4(5x-8)\cdot 5=20(5x-8)$

Logarithmic Differentiation:

This is a very useful method that can be used to find the derivative of any function.

Using the formula $\frac{d}{dx}(\ln u)=\frac{\frac{d}{dx}(u)}{u}$

We proceed as follows:

$y=f(x)$

$\ln y=\ln f(x)$

$(\ln y)'=(\ln f(x))'$

$\frac{y'}{y}=(\ln f(x))'$

$y'=y(\ln f(x))'$

$\frac{dy}{dx}=f(x)\cdot(\ln f(x))'$

Example:

Find the derivative of:

$y=\sqrt[3]{3x^{2}}$

Solution:

$\ln y=\ln (\sqrt[3]{3x^{2}})$

$\ln y=\frac{1}{3}\ln(3x^{2})$

$\frac{y'}{y}=\frac{1}{3}\frac{6x}{3x^{2}}=\frac{2}{3x}$

$y'=y\frac{2}{3x}=\frac{2}{3x}\sqrt[3]{3x^{2}}$

Second derivative

$f''=(f')'=\frac{d}{dx}(\frac{dy}{dx})=\frac{d^{y}}{dx^{2}}$

Higher-order derivatives

$f^{(n)}=\frac{d^{n}y}{dx^{n}}$

$(u+v)^{(n)}=u^{(n)}+v^{(n)}$

$(u-v)^{(n)}=u^{(n)}-v^{(n)}$

Leibnitz’s Formulas

$(uv)^{'''}=u'''v+3u''v'+3u'v''+v'''$

$(uv)^{(n)}=u^{(n)}v+nu^{(n-1)}v'+\frac{n(n-1)}{1\cdot 2}u^{(n-2)}v''+\cdots +uv^{(n)}$

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

$(x^{n})^{(n)}=n!$

$(\log_{a}x)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^{n} \ln a}$

$(\ln x)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^{n}}$

$(a^{x})^{(n)}=a^{x} \ln a$

$(e^{x})^{(n)}=e^{x}$

$(a^{mx})^{(n)}=m^{n}a^{mx}\ln^{n}a$

$(\cos x)^{(n)}=\cos(x+\frac{n \pi}{2})$

$(\sin x)^{(n)}=\sin(x+\frac{n \pi}{2})$

Some Applications of Derivative:

It will not be possible to list all applications of the derivative in few lines. However, through worked problems, this site will help you understand many aspects of the rate of change.

Remember that the derivative is the rate of change of some functions.

When you are driving a car, you start it, you accelerate and your speed increase at a certain rate. After some time, when you stop accelerating, you may drive at a constant speed. When you are near the point of arrival, you slow down and then stop.

The speed has increased, became constant and decreased. This is a typical derivative description. The rate of change was positive, so was the derivative during that period. Then the rate of change became $0$ and finally it became negative to the point of stop.

Mean value theorem:

If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then $f(b)-f(a)=f'(c)(b-a)$ for some number $c$ in $(a,b)$ .

Corollary :

If $f'(x)=0$ for all $x$ in $[a,b]$ , then $f$ is constant on $[a,b]$ , meaning that there is a constant $C$ such $f(x)=C$ for all $x$ in $[a,b]$

Corollary:

Let $f$ be a function that is continuous on $[a,b]$ and differentiable on $(a,b)$ :

If $f'(x)>0$ for all $x$ in $(a,b)$ , then $f$ is an INCRESAING FUNCTION on $[a,b]$

If $f'(x)<0$ for all $x$ in $(a,b)$ , then $f$ is an DECRESAING FUNCTION on $[a,b]$

First derivative Test Theorem:

Let the function $f$ be continuous on the open interval $(a,b)$ and be differentiable there except possibly at $c$

1. If $f'(x)<0$ on $(a,c)$ and $f'(x)>0$ on $(c,b)$ , then $f(c)$ is the minimum value of $f$ on $(a,b)$

2. If $f'(x)>0$ on $(a,c)$ and $f'(x)<0$ on $(c,b)$ , then $f(c)$ is the maximum value of $f$ on $(a,b)$

3. If $f'(x)>0$ or $f'(x)<0$ for all $x$ in $(a,b)$ except for $x=c$ , then $f(c)$ is neither a maximum nor a minimum value for $f$

Tangent line at a given point:

$y-y_{0}=f'(x_{0})(x-x_{0})$

Normal line at a given point:

$y-y_{0}=-\frac{1}{f'(x_{0})}(x-x_{0})$

Inflection Points

If $f'(x_{3})$ exists and $f''(x)$ changes sign at $x=x_{3}$ , the the point $(x_{3}, f(x_{3}))$ is an inflection point of the graph of $f(x)$ . If $f''(x_{3})$ exists at the inflection point, then $f''(x_{3})=0$