# 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) = 4x^{2} + 4

g(x) = x^{3} + 2

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

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

h(x) = x^{3} + 4x^{2} + 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) = 3x^{2} + 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) = 4x^{2} + 4

f’(x) = 8x

g(x) = x^{3} + 2

g’(x) = 3x^{2}

[f(x) + g(x)]’ = 3x^{2} + 8x

**Example:**

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

f(x) =8x^{4} + 6x^{3} + 7

g(x) = 5x^{3} + x^{2} + 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) = 32x^{3} + 18x^{2 }

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) = 15x^{2} + 2x + 1

Finally, let’s add these derivatives.

f’(x) + g’(x) = 32x^{3} + 18x^{2} + 15x^{2} + 2x + 1

f’(x) + g’(x) = 32x^{3} + 33x^{2}+ 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) = 7x^{6} + 5x^{4} + 8

g(x) = x^{4} + 2x^{3} + 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 (2x^{5})’ = 2•5•x^{4}

f(x) = 7x^{6} + 5x^{4} + 8

f’(x) = 7•6x^{(6-1)} + 5•4x^{(4 – 1)} + 0

f’(x) = 42x^{5} + 20x^{3}

g(x) = x^{4}+ 2x^{3} + 4x + 1

g’(x) = 4x^{(4 – 1)}+ 2•3x^{(3 – 1)} + 4

g’(x) = 4x^{3} + 6x^{2} + 4

f’(x) + g’(x) = 42x^{5} + 20x^{3} + 4x^{3} + 6x^{2} + 4

f’(x) + g’(x) = 42x^{5} + 20x^{3} + 4x^{3} + 6x^{2} + 4

**Example:**

A function f(x) is given as follows.

f(x) = | 1 |
| ||||||||||||||||

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) = -x^{70} + x^{69} - x^{68} + …. –x^{2} + 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•x^{69} + 69•x^{68} - 68•x^{67} + … - 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

- WRITE COMMENT
- NAME SURNAME(or nick)
- COMMENT