Find the derivative of f(x) = √x
Understand the Problem
The question is asking to find the derivative of the square root function, f(x) = √x. This can be solved using the power rule after rewriting the square root as a power.
Answer
$\frac{1}{2\sqrt{x}}$
Answer for screen readers
$f'(x) = \frac{1}{2\sqrt{x}}$
Steps to Solve
- Rewrite the square root as a power
We can rewrite the square root function $f(x) = \sqrt{x}$ as $f(x) = x^{\frac{1}{2}}$.
- Apply the power rule
The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Applying this rule to our function $f(x) = x^{\frac{1}{2}}$, we get:
$f'(x) = \frac{1}{2}x^{\frac{1}{2} - 1}$
- Simplify the exponent
Simplify the exponent $\frac{1}{2} - 1$:
$\frac{1}{2} - 1 = \frac{1}{2} - \frac{2}{2} = -\frac{1}{2}$
So, $f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$.
- Rewrite the negative exponent and simplify
Rewrite the term with the negative exponent as a fraction:
$f'(x) = \frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} = \frac{1}{2x^{\frac{1}{2}}}$
Finally, rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$:
$f'(x) = \frac{1}{2\sqrt{x}}$
$f'(x) = \frac{1}{2\sqrt{x}}$
More Information
The derivative of the square root function is a fundamental result in calculus. It's frequently used in optimization problems and related rates.
Tips
A common mistake is forgetting to subtract 1 from the exponent when applying the power rule. Another mistake could be incorrectly simplifying the fractional exponent. Also, some might forget how to rewrite fractional exponents as radicals and vice-versa.
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