Find the difference quotient by using

f(x+h)−f(x)/h

1) f(x) = √(2x+1)
answer

(√(2x+1+2h) - √(2x+1))/h

can you check please

correct so far, and answered previously

To find the difference quotient using the formula f(x+h) - f(x)/h, you need to substitute the function f(x) with the given function and simplify the expression.

In this case, the function is f(x) = √(2x + 1).

So, let's substitute f(x+h) and f(x) into the difference quotient formula:

f(x+h) - f(x) = √(2(x+h)+1) - √(2x+1)

Now, we divide the above expression by h:

(√(2(x+h)+1) - √(2x+1)) / h

This is the difference quotient. However, to simplify it further, we can perform some algebraic manipulation.

To eliminate the square roots in the numerator, we can multiply the expression by the conjugate of the numerator:

[(√(2(x+h)+1) - √(2x+1)) / h] * [(√(2(x+h)+1) + √(2x+1)) / (√(2(x+h)+1) + √(2x+1))]

Expanding and simplifying the numerator gives us:

[(2(x+h)+1) - (2x+1)] / [h * (√(2(x+h)+1) + √(2x+1))]

Simplifying further:

[2x + 2h + 1 - 2x - 1] / [h * (√(2(x+h)+1) + √(2x+1))]

Cancelling out like terms:

[2h] / [h * (√(2(x+h)+1) + √(2x+1))]

Now, we can divide the numerator and denominator by h:

[2] / [√(2(x+h)+1) + √(2x+1)]

Therefore, the difference quotient for f(x) = √(2x+1) is:

[2] / [√(2(x+h)+1) + √(2x+1)]