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.

  • {n}th Derivatives of Standard Functions

successive_formulas.jpg

 

  • 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}.

Advertisements
Posted in M I

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s