# Successive Differentiation

• ### The Concept of Differentiation

The derivative of a function ${f(x)}$ tells the rate of change of ${f(x)}$ w.r.t. ${x}$. More specifically, the derivative measures the sensitivity to change of a quantity ${y= f(x)}$, which is determined by another quantity ${x}$. ${x}$ is known as the independent variable and ${y}$ is known as the dependent variable.

Formally, the derivative is defined by the limit

${f'(x) = \frac {dy}{dx} = \lim \limits_{h \rightarrow 0} \frac {f(x+h)-f(x)}{h}}$

It finds its use in wide range of areas and is also linked to integration, which leads to the fundamental theorem of integral calculus.

• ### Successive Differentiation in Dynamics

Let the displacement ${\vec r}$, velocity ${\vec v}$ and acceleration ${\vec a}$ be functions of time ${t}$. By definition,

${\vec v = \frac {d \vec r}{dt}, \vec a = \frac {d \vec v}{dt} = \frac {d}{dt} \frac {d \vec r}{dt} = \frac {d^2 \vec r}{dt^2}}$

The jerk ${\vec j}$ is defined as the rate of change of acceleration. So,

${\vec j = \frac {d \vec a}{dt} = \frac {d}{dt} \frac {d^2 \vec r}{dt^2} = \frac {d^3 \vec r}{dt^3}}$

Thus, ${\vec v, \vec a, \vec j}$ are obtained by differentiating ${\vec r}$ successively.

• ### Successive Differentiation

Let ${y}$ be a function of ${x}$ only; ${y = f(x)}$. Then

${\frac {dy}{dx} = y_1}$ is the first derivative,

${\frac {d^2y}{dx^2} = y_2}$ is the second derivative,

${\frac {d^3y}{dx^3} = y_3}$ is the third derivative.

In general, ${\frac {d^ny}{dx^n} = y_n}$ is the ‘n’th derivative}.

• ### Infinitely Differentiable Functions

There are certain functions, whose derivatives exist for all values of ${n \in \mathbb N}$. These are known as infinitely differentiable functions. We are going to study these functions only.

• ### Important Points

I) Trigonometric Formulas

${sin (A+B) = sin(A)cos(B) \ + \ cos(A)sin(B)}$

${sin (A-B) = sin(A)cos(B) \ - \ cos(A)sin(B)}$

${cos (A+B) = cos(A)cos(B) \ - \ sin(A)sin(B)}$

${cos (A-B) = cos(A)cos(B) \ + \ sin(A)sin(B)}$

${tan (A+B) = \frac {tan(A)+tan(B)}{1- tan(A)tan(B)}}$

${tan (A-B) = \frac {tan(A)-tan(B)}{1+ tan(A)tan(B)}}$

II) Euler’s Identity

${re^{i \theta} = r [cos (\theta)+ i \ sin (\theta)],\ i = \sqrt {-1}}$

${cos (\theta) = Re [e^{i \theta}] \ and \ sin(\theta) = Im [e^{i \theta}]}$

III) Splitting into Partial Fractions

Consider ${\frac 1 {(x^2-3x+2)} = \frac {1}{(x-1)(x-2)} (= \ P \ say) = \frac {A}{(x-1)}+ \frac {B}{(x-2)}}$.

To get ${A}$, we solve ${x-1=0}$, put the value in ${P}$, except ${x-1}$. ${A = -1}$. Similarly, to get ${B}$, we solve ${x-2=0}$, and put the value in ${P}$, except ${x-2}$. ${B = 1}$.

These fractions can be easily differentiated by using ${(ax+b)^m}$ where ${m=-1}$.

IV) Combinations of ${n}$ objects, taken ${r}$ at a time, ${^n C_r}$ :

${^nC_r = \frac {n!}{r! (n-r)!}}$

• ### Leibniz Rule (Leibnitz Rule)

The rule is due to Leibniz, who invented calculus independently of Newton. His rule is useful for finding ${n}$th derivative of product of 2 functions. The rule says, if ${y=uv}$, then,

${y_n = ^nC_0 u_nv + ^nC_1 u_{n-1}v_1 + ^nC_2 u_{n-2}v_2 + \cdots + ^nC_r u_{n-r}v_r + \cdots + ^nC_n uv_{n}}$

• ### HINT FOR MCQs

In any problem asking for ${n}$th derivative of a function, get the first derivative of the function. Put ${n=1}$ in the options. See which of the options matches with the first derivative. If two or more options match, get the second derivative, put ${n=2}$ in the options and check.

Example : ${n}$th derivative of ${y=x^{n-1}}$ is :

I) ${(n-1)!}$,  II) ${n!x}$, III) ${n!}$,  IV) ${0}$

Put ${n=1}$, thus ${n-1 = 0}$, i.e. ${y=x^0 = 1}$, so first derivative will be ${0}$. Option ${IV}$.