Question

For x = (x-1)/(x+1) find the derivative at x=3 using the definition of derivative as a limit.

A) 3/16
B) 4/25
C) 2/9
D) 5/36

Answer 1

For x = (x-1)/(x+1), the value of the derivative at x=3 is
`\[ \text{Closest option:} \quad (-1)/(4) \]`

The correct option is not among the provided choices.

None of the choices A, B, C and D are correct

Explanation:

To find the derivative of
`\(f(x) = (x - 1)/(x + 1)\)` at x = 3 using the definition of the derivative as a limit, we'll use the following formula:

`\[ f'(x) = \lim_{{h \to 0}} (f(x + h) - f(x))/(h) \]`

First, find f(x):

`\[ f(x) = (x - 1)/(x + 1) \]`

Now, find f'(x) using the definition of the derivative:

`\[ f'(x) = \lim_{{h \to 0}} (((x + h) - 1)/((x + h) + 1) - (x - 1)/(x + 1))/(h) \]`

Combine the fractions:

`\[ f'(x) = \lim_{{h \to 0}} ((x + h - 1)/(x + h + 1) - (x - 1)/(x + 1))/(h) \]`

Combine the fractions in the numerator:

`\[ f'(x) = \lim_{{h \to 0}} (((x + h - 1)(x + 1) - (x - 1)(x + h + 1))/((x + h + 1)(x + 1)))/(h) \]`

Now, simplify the numerator:

`\[ f'(x) = \lim_{{h \to 0}} (x^2 + (2h - 2)x + h^2 - 1 - (x^2 - (2h + 2)x + h^2 + 1))/((x + h + 1)(x + 1)) \]`

Combine like terms:

`\[ f'(x) = \lim_{{h \to 0}} (4hx - 4)/((x + h + 1)(x + 1)) \]`

Now, substitute x = 3 into the expression:

`\[ f'(3) = \lim_{{h \to 0}} (12h - 4)/((3 + h + 1)(3 + 1)) \]`

Simplify further:

`\[ f'(3) = \lim_{{h \to 0}} (12h - 4)/((h + 4)(4)) \]`

Now, plug in h = 0:

`\[ f'(3) = (-4)/(4 \cdot 4) = (-1)/(4) \]`

So, the correct option is not among the provided choices. The closest option is:

`\[ \text{Closest option:} \quad (-1)/(4) \]`

None of the choices A, B, C and D are correct