# Derivative Of The Sum Of Functions

## Mathematics lesson, derivatives topic. Derivative of summing of two or more functions. The sum of finite derivatives of consecutive multiple terms. The rule of summing in derivatives.

The Rule Of Summing

If functions f and g are a function that from cluster A to the cluster A and if cluster A is subset of the real numbers cluster and if functions f and g have derivatives at point x ϵ A , derivatives of the sum of the functions f and g can found as follows.

[f(x) + g(x)]’ = f’(x) + g’(x)

let's assume that

f(x) = 4x2 + 4

g(x) = x3 + 2

And the function that occurs from the sum of these two functions is h(x)

h(x) = f(x) + g(x)

h(x) = x3 + 4x2 + 6

The function h(x) consists of three terms. To find the derivative of the function h(x), the derivative of each terms is taken separately.

h’(x) = 3x(3-1) + 8x(2-1) + 0

h‘(x) = 3x2 + 8x

This derivative can also be found is taken derivatives of the functions f(x) and g(x) separately and collecting these derivatives.

f(x) = 4x2 + 4

f’(x) = 8x

g(x) = x3 + 2

g’(x) = 3x2

[f(x) + g(x)]’ = 3x2 + 8x

Example:

Two functions that defined in real numbers cluster are given as below.

f(x) =8x4 + 6x3 + 7

g(x) = 5x3 + x2 + x + 6

Find the [f(x) + g(x)]’

Solution:

First, let's calculate the derivative of the function f(x).

f’(x) = 4•8x(4 – 1) + 6•3x(3 – 1) + 0

f’(x) = 32x3 + 18x

Now, let’s calculate the derivative of the function g(x).

g’(x) = 5•3x(3 – 1) + 2x(2 – 1)+ x(1 – 1) + 0

g’(x) = 15x2 + 2x + 1

f’(x) + g’(x) = 32x3 + 18x2 + 15x2 + 2x + 1

f’(x) + g’(x) = 32x3 + 33x2+ 2x + 1

Example:

The functions f(x) and g(x) are given as follows. Find derivative of the sum these functions.

Solution:   Example:

f(x) = 7x6 + 5x4 + 8

g(x) = x4 + 2x3 + 4x + 1

Find the [f(x) + g(x)]’

Solution:

When derivative of exponential numbers is taken, the coefficient is multiplied by the exponent and the exponent is decreased 1.

For example (2x5)’ = 2•5•x4

f(x) = 7x6 + 5x4 + 8

f’(x) = 7•6x(6-1) + 5•4x(4 – 1) + 0

f’(x) = 42x5 + 20x3

g(x) = x4+ 2x3 + 4x + 1

g’(x) = 4x(4 – 1)+ 2•3x(3 – 1) + 4

g’(x) = 4x3 + 6x2 + 4

f’(x) + g’(x) = 42x5 + 20x3 + 4x3 + 6x2 + 4

f’(x) + g’(x) = 42x5 + 20x3 + 4x3 + 6x2 + 4

Example:

A function f(x) is given as follows.

f(x) = 1
+1
+1
+ …+1
 + …+ 1 X90
X2
X3
X2
X

Find the value of the f’(1)

Solution:

We can write the function f(x) as following way.

f(x) = x-1 + x-2 + x-3 + …+ x-90

The derivative of this function.

f’(x) = -x -2 -2x-3 -3x-4 - ….-90x-91

The value of the derivative of this function at point x = 1 is

f’(1) = -1-2 – 2•1-3 – 3•1-4 - …-90•1-91

f’(1) = -1 – 2 – 3 - …. – 90

The sum of numbers from -1 to -90 is

 - (90 • 91) = – 4095 2

Example:

f(x) = -x70 + x69 - x68 + …. –x2 + x

Find the derivative of the function f(x) and value of the f’(-1)

Solution:

The derivative of the function f(x) is

f’(x) = -70•x69 + 69•x68 - 68•x67 + … - 2•x + 1

For x = -1,

f’(-1) = - 70(-1)69 + 69(-1)68 - 68(-1)67 + … -2(-1)1 + 1

f’(1) = 70 + 69 + 68 + …+ 2 + 1

The sum of the consecutive numbers from 1 to 70 is

 70 • 71 = 2485 2

f’(-1) = 2485

Multiplication Rule In Derivatives

Derivative Of Parametric Functions

RISE KNOWLEDGE

February 12 2019

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