Question

Describe the x-values at which the function is differentiable. (Enter your answer using interval notation.)

y =x^2/x² - 49

Answer 1

Answer: The x-values at which the function is differentiable are given by the open interval (-∞,-7)U(-7,7)U(7,∞)

Explanation:

Given:
`y=x^(2) /x^(2) -49`

To Find: The x- values at which the function is differentiable.

Solution: First , We need to check whether the function is continuous and well- defined derivative at every point within its domain.

The given function is,

It is clear from the given function that the denominator
`x^(2) -49\\eq 0` .

Therefore,
`x\\eq` ±7 .

Also , we can simply the function as follows:

`y=x^(2) /(x-7)(x+7)` - (1)

Now , Differentiate the above Equation with respect to x by using u/v method ( Quotient Rule),

`y' = { (x-7)(x+7)(2x) - x^(2) [ (x+7) +(x-7)] } / (x-7)^(2) (x+7)^(2)`

`y' =( x^(2) -49)2x - x^(2) (2x)/(x-7)^(2) (x+7)^(2)`

`y'= 2x^(3) -98x-2x^(3) / (x-7)^(2) (x+7)^(2)`

`y'= -98x/(x-7)^(2)(x+7)^(2)`

We note that ,

y' is defined for all x except x = ± 7 , which are the points of discontinuity of the original function.

Thus , the x-values at which the function is differentiable are given by the open interval (-∞,-7)U(-7,7)U(7,∞)

Answer:

x-values at which the function is differentiable in interval notation is ,

(-∞,-7)U(-7,7)U(7,∞)