derivative of square root x
Understand the Problem
The question is asking for the derivative of the function square root of x. To solve this, we will apply the rules of differentiation, specifically the power rule, since the square root can be expressed as x raised to the power of 1/2.
Answer
\( \frac{1}{2 \sqrt{x}} \)
Answer for screen readers
The final answer is ( \frac{1}{2 \sqrt{x}} )
Steps to Solve
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Rewrite the square root function using exponentiation
The square root of $x$ can be expressed as an exponent:
$$ \sqrt{x} = x^{1/2} $$
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Apply the power rule for differentiation
The power rule states that the derivative of $x^n$ is $nx^{n-1}$.
For our function $x^{1/2}$, apply the power rule:
$$ \frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} x^{(1/2) - 1} $$
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Simplify the exponent
Simplify the exponent $\frac{1}{2} - 1$:
$$ \frac{1}{2} - 1 = -\frac{1}{2} $$
So, the expression becomes:
$$ \frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} x^{-1/2} $$
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Rewrite to standard form
Rewrite $x^{-1/2}$ as $\frac{1}{\sqrt{x}}$:
$$ \frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} \cdot \frac{1}{\sqrt{x}} = \frac{1}{2\sqrt{x}} $$
The final answer is ( \frac{1}{2 \sqrt{x}} )
More Information
The process used to find the derivative utilizes the power rule of differentiation, which is a fundamental concept in calculus. This derivative is useful in many applications including physics and engineering.
Tips
A common mistake is forgetting to adjust the exponent correctly when applying the power rule. Always ensure to subtract 1 from the given exponent.
