Question

Help me find this asap

Answer 1

DQ =
`(5)/((2-√(x))(2-√(x+h))(√(x)-√(x+h)))`

Explanation:

Given function is f(x) =
`(5)/(2-√(x))`

f(x + h) =
`(5)/(2-√(x+h) )`

Therefore, indicated difference quotient will be,

DQ =
`(f(x+h)-f(x))/(h)`

Now we substitute the values in the difference quotient,

DQ =
`((5)/(2-√(x+h) )-(5)/(2-√(x)))/(h)`

=
`(5(2-√(x))-5(2-√(x+h)))/(h(2-√(x))(2-√(x+h)))`

=
`(10-5√(x)-10+5√(x+h))/(h(2-√(x))(2-√(x+h)))`

=
`(-5√(x)+5√(x+h))/(h(2-√(x))(2-√(x+h)))`

=
`(5(-√(x)+√(x+h)))/(h(2-√(x))(2-√(x+h)))`

=
`(5(-√(x)+√(x+h))(√(x)+√(x+h)))/(h(2-√(x))(2-√(x+h))(√(x)+√(x+h)))`

=
`(5[(x+h)-x])/(h(2-√(x))(2-√(x+h))(√(x)+√(x+h)))`

=
`(5h)/(h(2-√(x))(2-√(x+h))(√(x)+√(x+h)))`

=
`(5)/((2-√(x))(2-√(x+h))(√(x)+√(x+h)))`