Answer:
Step-by-step explanation:
So we have the function:
And we want to find the derivative using the limit process.
The definition of a derivative as a limit is:
Therefore, our derivative would be:
First of all, let's factor out a 4 from the numerator and place it in front of our limit:
Place the 4 in front:
Now, let's multiply everything by (√(x+h)(√(x))) to get rid of the fractions in the denominator. Therefore:
Distribute:
Simplify: For the first term on the left, the √(x+h) cancels. For the term on the right, the (√(x)) cancel. Thus:
Now, multiply both sides by the conjugate of the numerator. In other words, multiply by (√x + √(x+h)). Thus:
The numerator will use the difference of two squares. Thus:
Simplify the numerator:
Both the numerator and denominator have a h. Cancel them:
Now, substitute 0 for h. So:
Simplify:
(√x)(√x) is just x. (√x)+(√x) is just 2(√x). Therefore:
Multiply across:
Reduce. Change √x to x^(1/2). So:
Add the exponents:
And we're done!