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- Numerical Differentiation
- Differentiation Formulas
- Derivatives of Inverse Trig Functions
- Differentiation by Parts
- Derivative of Implicit Functions
- Logarithmic Differentiation
- Differential Equation
- Solving First Order Differential Equation
- Homogeneous Differential Equation
- Derivatives Practice Problems

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.

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.

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

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

2. Derivative of ( e

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

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

5. Derivative of ( a

(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 sec

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

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

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

(ii) Derivative of cos

(iii) Derivative of tan

(iv) Derivative of cot

(v) Derivative of sec

(vi) Derivative of csc

$\frac{dy}{dx}$(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

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 $\frac{d\Psi}{dx}$* $(\frac{dy}{dx})$.

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

If a differential equation contains only one independent variable, then it is called an

The exponent of x and y need not be an integer.

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’

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.

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.

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’

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

Step 1:

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

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