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3 years ago

What is the diffrentiation of logx

Mathematics

1 answer:

dalvyx [7]3 years ago

5 0

Derivative of Logarithm

When the logarithmic function is given by:

f (x) = logb(x)

The derivative of the logarithmic function is given by:

f ' (x) = 1 / ( x ln(b) )

x is the function argument.

b is the logarithm base.

ln b is the natural logarithm of b.

For example when:

f (x) = log2(x)

f ' (x) = 1 / ( x ln(2) )

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Let C(n, k) = the number of k-membered subsets of an n-membered set. Find (a) C(6, k) for k = 0,1,2,...,6 (b) C(7, k) for k = 0,

vladimir1956 [14]

Answer:

(a) $C(6,0) = 1$ , $C(6,1) = 6$ , $C(6,2) = 15$ , $C(6,3) = 20$ , $C(6,4) = 15$ , $C(6,5) = 6$ , $C(6,6) = 1$ .

(b) $C(7,0) = 1$ , $C(7,1) = 7$ , $C(7,2) = 21$ , $C(7,3) = 35$ , $C(7,4) = 35$ , $C(7,5) = 21$ , $C(7,6) = 7$ , $C(7,7)=1$ .

Step-by-step explanation:

In this exercise we only need to recall the formula for C(n,k):

$C(n,k) = \frac{n!}{k!(n-k)!}$

where the symbol $n!$ is the factorial and means

$n! = 1\cdot 2\cdot 3\cdot 4\cdtos (n-1)\cdot n$ .

By convention 0!=1. The most important property of the factorial is $n!=(n-1)!\cdot n$ , for example 3!=1*2*3=6.

(a) The explanations to the solutions is just the calculations.

$C(6,0) = \frac{6!}{0!(6-0)!} = \frac{6!}{6!} = 1$

$C(6,1) = \frac{6!}{1!(6-1)!} = \frac{6!}{5!} = \frac{5!\cdot 6}{5!} = 6$

$C(6,2) = \frac{6!}{2!(6-2)!} = \frac{6!}{2\cdot 4!} = \frac{5!\cdot 6}{2\cdot 4!} = \frac{4!\cdot 5\cdot 6}{2\cdot 4!} = \frac{5\cdot 6}{2} = 15$

$C(6,3) = \frac{6!}{3!(6-3)!} = \frac{6!}{3!\cdot 3!} = \frac{5!\cdot 6}{6\cdot 6} = \frac{5!}{6} = \frac{120}{6} = 20$

$C(6,4) = \frac{6!}{4!(6-4)!} = \frac{6!}{4!\cdot 2!} = frac{5!\cdot 6}{2\cdot 4!} = \frac{4!\cdot 5\cdot 6}{2\cdot 4!} = \frac{5\cdot 6}{2} = 15$

$C(6,5) = \frac{6!}{5!(6-5)!} = \frac{6!}{5!} = \frac{5!\cdot 6}{5!} = 6$

$C(6,6) = \frac{6!}{6!(6-6)!} = \frac{6!}{6!} = 1$ .

(b) The explanations to the solutions is just the calculations.

$C(7,0) = \frac{7!}{0!(7-0)!} = \frac{7!}{7!} = 1$

$C(7,1) = \frac{7!}{1!(7-1)!} = \frac{7!}{6!} = \frac{6!\cdot 7}{6!} = 7$

$C(7,2) = \frac{7!}{2!(7-2)!} = \frac{7!}{2\cdot 5!} = \frac{6!\cdot 7}{2\cdot 5!} = \frac{5!\cdot 6\cdot 7}{2\cdot 5!} = \frac{6\cdot 7}{2} = 21$

$C(7,3) = \frac{7!}{3!(7-3)!} = \frac{7!}{3!\cdot 4!} = \frac{6!\cdot 7}{6\cdot 4!} = \frac{5!\cdot 6\cdot 7}{6\cdot 4!} = \frac{120\cdot 7}{24} = 35$

$C(7,4) = \frac{7!}{4!(7-4)!} = \frac{6!\cdot 7}{4!\cdot 3!} = frac{5!\cdot 6\cdot 7}{4!\cdot 6} = \frac{120\cdot 7}{24} = 35$

$C(7,5) = \frac{7!}{5!(7-2)!} = \frac{7!}{5!\cdot 2!} = 21$

$C(7,6) = \frac{7!}{6!(7-6)!} = \frac{7!}{6!} = \frac{6!\cdot 7}{6!} = 7$

$C(7,7) = \frac{7!}{7!(7-7)!} = \frac{7!}{7!} = 1$

For all the calculations just recall that 4! =24 and 5!=120.

6 0

3 years ago

What is the sum of a geometric sequence 1,3,9,... if there are 14 terms

Maksim231197 [3]

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323.
If this is the proper sequence pattern, we add up all the numbers to get a sum.
The total sum is 2391457.

7 0

3 years ago

I need some help on these 3 questions :)

wlad13 [49]

The factor of the polynomial, the area of the pool and the simplified expression are all algebraic expressions

<h3>How to simplify the expressions?</h3>

Factor of the polynomial

The polynomial is given as:

2x^3 + 6x^2 + 6x + 18

Factor out 2

2x^3 + 6x^2 + 6x + 18 = 2(x^3 + 3x^2 + 3x + 9)

Factorize

2x^3 + 6x^2 + 6x + 18 = 2(x^2(x + 3) + 3(x + 3))

Factor out x + 3

2x^3 + 6x^2 + 6x + 18 = 2(x^2 + 3)(x + 3)

Expand

2x^3 + 6x^2 + 6x + 18 = (2x^2 + 6)(x + 3)

Hence, 2x^2 + 6 is a factor of the polynomial 2x^3 + 6x^2 + 6x + 18

The length of the pool

The given parameters are:

Area = 2x^3 - 29x + 12

Width = x + 4

The length is calculated as:

Length = Area/Width

This gives

Length = 2x^2 - 29x + 12/x + 4

Factorize the numerator

Length = (2x^2 - 8x + 3)(x + 4)/x + 4

Cancel out x + 4

Length = 2x^2 - 8x + 3

Hence, the length of the pool is 2x^2 - 8x + 3

Simplify the expression

The expression is given as:

(3s^2 - 2s + 1) $\div$ (s^2 - s + 2)

Evaluate the quotient

3 + (s - 5)/(s^2 - 5 + 2)

Hence, the simplified expression is 3 + (s - 5)/(s^2 - 5 + 2)

Read more about algebraic expressions at:

brainly.com/question/2164351

3 0

2 years ago

Show me the break down for 52divide into 5256

Blizzard [7]

It is in all 0.0098934551 in all in total

4 0

3 years ago

Is 1/8 x 4= 1/12 or 4/8

aliina [53]

Answer:

If you were to fully simplify it would technically be 1/2 as 4/8 is correct but can be further rounded down

Step-by-step explanation:

1/8 x 4 = 4/8

4/8 ÷ 4 = 1/2

5 0

2 years ago

Read 2 more answers