How do you differentiate the square root function f(x) = √(1 + x)?

Understanding the Square Root Function

The square root function in question is f(x) = √(1 + x). To differentiate this function, we’ll use the rules of calculus, specifically the chain rule.

Step-by-Step Differentiation

1. Identify the outer and inner functions: In this case, the outer function is the square root, and the inner function is g(x) = 1 + x.
2. Differentiate the outer function: The derivative of √u (where u = g(x)) is 1/(2√u). Thus, the derivative of √(1 + x) becomes:

3. f'(x) = 1/(2√(1 + x))

Using the Chain Rule

Now, we need to multiply this by the derivative of the inner function g(x) = 1 + x, which is simply:

g'(x) = 1

Combining both derivatives, we find that:

f'(x) = 1/(2√(1 + x)) * 1 = 1/(2√(1 + x))

Final Result

Therefore, the derivative of the function f(x) = √(1 + x) is:

f'(x) = 1/(2√(1 + x))

Conclusion

In summary, the process of differentiating the square root function employs the chain rule effectively. Understanding how to apply these principles is crucial for more advanced calculus concepts. As always, practice will deepen your understanding, so don’t hesitate to work through more examples!