Differentiation

An important tools that is used in differential calculus is the derivative of a function.
Let us assume the given function is denoted as y = f(x). If the independent variable changes from the value ‘x’ to some other value x + $\partial $ x then $\partial $x is called an increment of ‘x’. Similarly $\partial $y denotes an increment in ‘y’. If ‘x’ changes x + x then the function y = f(x) will change from ‘y’ to y + $\partial $y so that


        y +$\partial $ y = f( x +$\partial $ x )
            $\partial $ y = f( x + $\partial $x ) – y
            $\partial $y = f( $\partial $x + x) – f(x)
Divide both sides with Δx

                 $\frac{\partial y}{\partial x}$ = $\frac{f(x+\partial k)-f(x)}{\partial k}$

Applying limit $\partial $ x tends to ‘0’ on both sides and
If we denote
         $\lim_{\partial x \rightarrow 0}$ $\frac{\partial y}{\partial x}$=0

        $\frac{dy}{dx}$=$\lim_{\partial x \rightarrow 0}$$\frac{f(x+\partial k)-f(x)}{\partial k}$

$\frac{dy}{dx}$ is called derivative of ‘y’ with respect to ‘x’. The derivative of f(x) is denoted as f’(x), D’.

Differentiation is the process of finding the derivative of a function.

Numerical Differentiation

The method that is used to estimate the derivative of a function is known as Numerical Differentiation.

The simplest way to estimate the derivative of a function is given by the formula

If f ( x ) is a mathematical function, then it’s derivative is given by

    f(x)=$\lim_{h\rightarrow 0}$ $\frac{f(x+h)-f(x)}{h}$

Differentiation Formulas

1.  Derivative of ( xn ) = n xn – 1

2. Derivative of ( ex ) = ex

3. Derivative of log( x ) = $\frac{1}{x}$

4. Derivative of log( x ) to the base ‘a’ = $\frac{log a}{x}$

5. Derivative of ( ax ) = ax * ln( a )


List of derivatives of trigonometric functions

(i) Derivative of sin ($\theta$) is equal to cos ($\theta$)

(ii) Derivative of cos ($\theta$) is equal to - sin ($\theta$)

(iii) Derivative of tan ($\theta$) is equal to sec2 ($\theta$)

(iv) Derivative of cot ($\theta$) is equal to – csc2 ($\theta$)

(v) Derivative of sec ($\theta$) is equal to sec ($\theta$) . tan ($\theta$ )

(vi) Derivative of csc ($\theta$) is equal to - csc ($\theta$) . cot ($\theta$)

Derivatives of Inverse Trig Functions

(i) Derivative of sin-1 ( u ) is equal to $\frac{1}{\sqrt{(1 – u^{2})}}$    or    $\frac{1}{(1 – u^{2} )^{\frac{1}{2}}}$

(ii) Derivative of cos-1 (u ) is equal to $\frac{- 1}{\sqrt{(1 – u^{2})}}$    or    $\frac{-1}{(1 – u^{2} )^{\frac{1}{2}}}$

(iii) Derivative of tan-1 ( u ) is equal to $\frac{1}{(1 + u^{2})}$

(iv) Derivative of cot-1 ( u ) is equal to $\frac{-1}{(1 + u^{2})}$

(v) Derivative of sec-1 (u ) is equal to $\frac{1}{[|u| * \sqrt{(u^{2} - 1)}]}$    or    $\frac{1}{[|u| (u^{2} - 1)^{\frac{1}{2}}]}$

(vi) Derivative of csc-1 (u) is equal to $\frac{-1}{[|u| * \sqrt{(u^{2} - 1)}]}$    or    $\frac{-1}{[|u| (u^{2} - 1)^{\frac{1}{2}}]}$

Differentiation by Parts

If ‘u(x)’ and ‘v(x)’ are differentiable functions of ‘x’ then the product ‘u * v’ is also differentiable such that

$\frac{dy}{dx}$(uv) = u $\frac{dv}{dx}$ (v) + v
$\frac{dv}{dx}$ (u)

Let us see a worked out example that will clearly explain the concept of the product rule for derivatives

Differentiation by parts Example

Derivative of Implicit Functions

If the variable ‘x’ and ‘y’ are connected by a relation of the form f(x, y) = 0 and it is not possible or convenient to express ‘y’ as a function of ‘x’ in the form y $\Psi $= (x), then ‘y’ is said to be an implicit function of ‘x’.

To find the derivative of ‘y’ in such a case, we differentiate both sides of the given relation with respect to ‘x’, keeping in mind that the derivative of (y) with respect to ‘x’ is $\frac{d\Psi}{dx}$* $(\frac{dy}{dx})$.

Example of Implicit Differentiation:

Logarithmic Differentiation

We have seen derivatives of the functions of the form [f (x) ]n, nf(x) where f(x) is a function of ‘ x ‘ and ‘ n

‘is a constant. Here we shall mainly discuss derivatives of the functions that are in the form of [f(x)]g(x) 

where f ( x ) and g ( x ) are functions of ‘x’. To find the derivative of this type of functions we proceed by

applying logarithm on both sides and then differentiate.

Let us understand the concept of logarithmic differentiation by the following examples

Example of Logarithmic Differentiation:

Differential Equation

An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a differential equation.

If a differential equation contains only one independent variable, then it is called an Ordinary differential equation and if it contains more than one independent variable then it is called a Partial differential equation.

The exponent of x and y need not be an integer.
Order of a Differential Equation:
The Order of a differential equation is the order of the highest derivative occurring in it.

First Order Differential Equation

A first degree first order differential equation contains terms like dy/dx and some terms involving ‘x’ and ‘y’, a general first degree first order differential equation is of the form
           
$\frac{dy}{dx}$= F ( x, y ), where F is a function of ‘x’ and ‘y’


Solving First Order Differential Equation

If the given first order differential equation can be put in the form

            f (x ) dx + g ( y ) dy = 0
Then its solution can be obtained by integrating each term. This method of solving the first order differential equation is called variable separable method.


First Order Differential Equation Example:

Solution: We are given that, x dy – y dx = 0.

This can be written as x dy = y dx

 $\left(\frac{dx}{x} \right )$ = $\left (\frac{dy}{y} \right )$

On integrating both sides with respect to ‘ x ‘, we get


        log (x) = log (y) + log ( c )

        log (x) = log ( yc ), where ‘ c ‘ being an arbitrary positive constant.

That is, x = yc is the required solution of the given first order differential equation.


Second Order Differential Equation

A second order differential equation will be of the form

    f(x)$\frac{d^y}{dx^2}$ + G(x) $\frac{dy}{dx}$ + H(x) y = P(x)

Where F ( x ), G ( x ), H ( x ) and P ( x ) are continuous functions in ‘x’

Homogeneous Differential Equation

Definition: A function f ( x, y ) is called a homogeneous function of degree ‘ n ‘, if

            f ( cx, cy ) = cn f(x, y)

Definition of a homogeneous differential equation   
A first order first degree differential equation is expressible in the form

$\frac{dy}{dx}$ = $\frac{f(x,y)}{g(x,y)}$       
Where f ( x, y ) and g ( x, y ) are homogeneous functions of the same degree, then it is called a Homogeneous Differential Equation

Let us write an algorithm to solve a Homogeneous Differential equation
Where f ( x, y ) and g ( x, y ) are homogeneous functions of the same degree, then it is called a Homogeneous Differential Equation

Let us write an algorithm to solve a Homogeneous Differential equation

Algorithm:

Step 1:
Put the given differential equation in the form of

$\frac{dy}{dx}$ = $\frac{f(x,y)}{g(x,y)}$ 

Step 2: Put y = vx and $\frac{dy}{dx}$  = v + x*$\frac{dy}{dx}$  in the equation in step 1 and cancel out ‘x’ from the right hand side. Then the equation reduces to the form

        v + x*$\frac{dy}{dx}$  = F ( v )

Step 3: Shift ‘v’ on right hand side and separate the variables ‘v’ and ‘x’

Step 4: Integrate both sides to obtain the solution in terms of ‘v’ and ‘x’

Step 5: Replace ‘v’ by ($\frac{y}{x}$) in the solution obtained in step 4 to obtain the solution in terms of ‘x’ and ‘y’.

Derivatives Practice Problems