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What is a derivative?

A derivative represents the change of one variable with respect to the other. Let us now learn the various application of derivative formulas.

The derivative of a function f at a point x, denoted by f ′(x), is

Provided that the limit exists.

For example: Let  f (x) = x5 + 6x , find f ′(a).

The Derivative as a Function

The derivative of a function f (x) is just another function of x and hence is not necessarily independent of x. There can be various forms of the function, that is, it can be in the form of a linear equation, quadratic equation, trigonometric equation, hyperbolic equation, logarithmic equation and many more. All of them follow their own laws of differentiation.

Definition:  The derivative of a function f (x) is the function f ′ (x) and can be written as

For all x for which this limit exists. You have to note that here, h tends to 0.

Basic Differentiation Formulas

Suppose f (x) and g (x) are differentiable functions, c is any real number, then

  • Differentiating a Constant value

Where c is a constant

For example.

In simpler terms, any function independent of x that is with only a constant component when differentiated equals 0.

  • Addition function derivative

When differentiating a function which can be broken down into 2 simpler functions, in the same variable, the easiest method to differentiate them is to do it separately and then adding them up to simplify them further.

For example,

= 2

  • Subtraction function derivative

When a function is given such that it can be transformed into 2 separate equations where one is being subtracted from the other, you may follow the rule mentioned below. You must be extremely careful of the negative sign while following this rule.

  • Constant and a function derivative

When a constant value and a function are placed alongside each other, you simply have to differentiate the constant and multiply it with the constant. In simpler terms, you do not differentiate the constant.

  • Multiplication Function derivative

When a function can be broken down into two separate functions say f(x) and g(x) in a multiplicative form, you first multiply f(x) while keeping g(x) constant and then multiply g(x) while keeping f(x) constant. You then add them both to get the result.

 

  • Division function derivative

The rule to be followed in the division function is given below. You must be extremely careful of the signs and powers when working on a division derivative.

The Exponential Rule

 

The Power Rule

For any number n, which is real

Some basic examples for better understanding

For example: Differentiate the following

On further implication, you arrive at the answer.

Natural Logarithmic Rule

Trigonometric Rules

Derivative of inverse functions

x(y) is the inverse of the function y(x),

dy/dx = 1/dx/dy

Hyperbolic functions

  • d/dx ( sinhx ) = coshx
  • d/dx ( coshx) = sinhx
  • d/dx ( tanhx ) = sech2x
  • d/dx ( cothx ) = – cosech2x
  • d/dx ( sechx ) = – sechx. tanhx
  • d/dx ( cosechx ) = -cosechx . cothx

Chain Rule

Example, f (x) = g7 and g (x) = (x+3)

Then d/dx f(x) = d/dx (g7) d/dx (x+3)

=7(x+3)6 (1)

These are the basic derivative formulas that once mastered will help you solve questions very easily.

Now lets solve a few questions of Derivative Formulas

  1. d/dx [ 3x3 ] = d/dx [ 3x3 ] = 3 d/dx (x3) = 3.3 x3-1 = 9.x2
  2. d/dx ( 3x2 + 2x ) = d/dx (3x2) +d/dx (2x) = 3 d/dx (x2) +2 d/dx (x) = 6x + 2
  3. d/dx [ (2x+1)(x-4) ] = ( 2x+1 ) d/dx ( x-4 ) +( x-4) d/dx ( 2x+1 ) = ( 2x+1 )( 1) + ( x-4 )(2) = 2x + 1 + 2x -8 = 4x – 7
  4. Find dy / dx for

  1. Find the fifth derivative of f (x) = 2x4 – 3x3 + 5x2 – x – 1

  1. Find dy/dx for x4/5

y = x 4/5

dy / dx = (4/5) x(4/5) – 1 = (4/5)/ x1/5

  1. Find derivative of f (x) = x2cosx + sinx

dy/dx = x2 ( -sinx ) + 2x cosx + cos x
= -x2 sinx + 2x cosx + cos x

Lets take a test

  1. Find derivative of the following function:

F ( x ) = (x5 – 3x)(1/x2)

Answer: 3x2+3/x2

  1. Find the value of the function f ( x ) = ( x3 – 2 )2

Answer: 6x2 (x3 – 2)

  1. Find the value of the derivative of the following function 

Answer: -5 / ( 2x – 3)2

  1. Find the derivative of the function 

Answer:

Dy/dy = 4.9 at 

Hint: use the quotient rule

  1. Find the value of the derivative of y = x sinx + 2
    Answer: x cosx + sinx
  2. Find the value of the derivative of y = x2 cosx – x tanx – 1
    Answer: 2x cosx – x2 sinx – tanx – x sec2x
  3. Find the value of the derivative of y = 1 + cosx / 1 – cosx

Answer: – 2sinx / ( 1 – cosx )2

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