Differentiate a square root.
Understand the Problem
The question is asking for the differentiation of a function involving a square root. Specifically, we will apply the rules of differentiation to find the derivative of a square root function.
Answer
The derivative of $f(x) = \sqrt{x}$ is $f'(x) = \frac{1}{2\sqrt{x}}$.
Answer for screen readers
The derivative of the function $f(x) = \sqrt{x}$ is given by:
$$ f'(x) = \frac{1}{2\sqrt{x}} $$
Steps to Solve
- Identify the function to differentiate
We will start by identifying the function. For example, if our function is $f(x) = \sqrt{x}$, we need to express this in a suitable form for differentiation.
- Rewrite the square root in exponent form
The square root function can be rewritten using exponents. We express $f(x) = \sqrt{x}$ as:
$$ f(x) = x^{1/2} $$
- Apply the power rule of differentiation
Now we will use the power rule for differentiation, which states that if $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$.
Applying this rule to our function:
$$ f'(x) = \frac{1}{2} \cdot x^{1/2 - 1} $$
- Simplify the derivative
Next, we simplify the derivative obtained in the previous step:
$$ f'(x) = \frac{1}{2} \cdot x^{-1/2} $$
This can also be written as:
$$ f'(x) = \frac{1}{2\sqrt{x}} $$
- State the final answer
We now state the derivative clearly based on our calculations.
The derivative of the function $f(x) = \sqrt{x}$ is given by:
$$ f'(x) = \frac{1}{2\sqrt{x}} $$
More Information
The derivative tells us the rate of change of the function $f(x) = \sqrt{x}$. This derivative is widely used in calculus, especially in problems related to optimization and motion.
Tips
- Confusing the power rule: Remember that when applying the power rule, you must decrease the exponent by one.
- Forgetting to simplify the expression: It’s important to express the derivative in the simplest form for easy interpretation.
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