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


Finally, let’s add these derivatives.

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.

derivativesum_s1i1

Find derivative of the sum these functions.


Solution:

derivativesum_s1i2

derivativesum_s1i3


derivativesum_s1i4


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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