# Determine f'(1) , f(x)=(x+5)/(2x+1).

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Given:

f(x) = (x + 5)/(2x +1)

Let:

u = x + 5

v = 2x + 1

Then:

f(x) = u/v

And:

f'(x) = (u'v - uv')/v^2

u' = du/dx = 1

v' = dv/dx = 2

Substituting values of u, v, u, and v' in equation for f'(x):

f'(x) = [(2x + 1) - 2(x + 5)]/[(2x + 1)^2]

= 2x + 1 - 2x - 10)/[(2x + 1)^2]

= -9/[(2x + 1)^2]

Therefore:

f'(1) = -9/[(2*1 +1)^2]

= -9/9 = -1

For any three functions where f(x)=g(x)/h(x),

f'(x)= [h( x )* g'( x ) - g( x )* h'( x ) ] / [h( x )]^2

Here g(x)= x+5, g'(x)= 1

h(x)=2x+1, h'(x)=2

Now substituting these values in the expression for f'(x), we get

f'(x)=[(2x+1)*1-(x+5)*2]/((2x+1)^2

As we need the value of f'(1) there is no point of solving the above expression. Instead, directly substitute x=1.

We get f'(x)=[(2x+1)*1-(x+5)*2]/((2x+1)^2= [(2+1)-6*2]/(2+1)^2

=(3-12)/9

=-9/9

=-1

**Therefore f'(x)=-1**

f(x) = (x+5)/(2x+1)

To find f'(1).

Solution :

We shal find f'(x) first and then f'(1).

f(x) = (x+5)/(2x+1) = {(x+1/2) + 5- 1/2)}/(2x+1) = (1/2) + (4+1/2)/(2x+1)

f(x) = (1/2) + 4.5/(2x+1)

Taking differential coefficient, we get:

f'(x) = (1/2)' +4.5 {2x+1)^-1}'

f'(x) = 0 + (-1)(2x+1)^(-1-1) * (2x)', as (x^n )' = n x^(n-1). and (constant)' = 0

f'(x) = -{(2x+1)^(-2)}(2)

f'(1) = -2 (2*1+1)^(-2)

f'(1) = -2/(3^2) = -2/9

To calculate f'(1), we'll use the definition of the derivative:

f'(1) = lim {[f(x) - f(1)]/(x-1)} , x-->1

f(1) = (1+5)/(2+1) = 6/3 = 2

f'(1) = lim {[(x+5)/(2x+1) - 2]/ (x-1)}, x-->1

f'(1) = lim (x + 5 - 4x - 2) / (x-1)(2x+1), x-->1

We'll combine like terms from numerator:

f'(1) = lim (-3x + 3)/ (x-1)(2x+1), x-->1

We'll factorize by -3:

f'(1) = lim -3(x - 1)/ (x - 1)(2x+1), x-->1

We'll reduce like terms:

f'(1) = lim -3 / (2x+1), x-->1

We'll substitute x by 1:

f'(1) = -3 / (2+1)

f'(1) = -3/3

**f'(1) = -1**

We could also use the rule of quotient;

(u/v)' = (u'*v - u*v')/v^2

u = (x+5) => u' = 1

v = (2x+1) => v' = 2

(u/v)' = [(2x+1) - 2*(x+5)]/(2x+1)^2

We'll remove the brackets:

(u/v)' = (2x+1 - 2x-10)/(2x+1)^2

We'll eliminate like terms:

f'(x) = (u/v)' = -9/(2x+1)^2

We'll calculate f'(1):

f'(1) = -9/(2*1+1)^2

f'(1) = -9/3^2

f'(1) = -9/9

**f'(1) = -1**