Find the derivative of 1/√x.
Understand the Problem
The question is asking for the derivative of the function 1/√x with respect to x. To find this, we can use the rules of differentiation, specifically the power rule after rewriting the expression in terms of exponents.
Answer
The derivative is $ f'(x) = -\frac{1}{2\sqrt{x^3}} $.
Answer for screen readers
The derivative of the function $\frac{1}{\sqrt{x}}$ with respect to $x$ is
$$ f'(x) = -\frac{1}{2\sqrt{x^3}} $$
Steps to Solve
- Rewrite the Function
We start by rewriting the function in terms of exponents to make differentiation easier.
The function $f(x) = \frac{1}{\sqrt{x}}$ can be rewritten as:
$$ f(x) = x^{-\frac{1}{2}} $$
- Apply the Power Rule
Now, we apply the power rule of differentiation. The power rule states that if $f(x) = x^n$, then the derivative $f'(x) = n \cdot x^{n-1}$.
In our case, the exponent $n = -\frac{1}{2}$, so we differentiate:
$$ f'(x) = -\frac{1}{2} \cdot x^{-\frac{1}{2} - 1} $$
- Simplify the Derivative
Now we simplify the expression:
$$ -\frac{1}{2} \cdot x^{-\frac{3}{2}} $$
This can also be rewritten back in terms of a square root:
$$ f'(x) = -\frac{1}{2\sqrt{x^3}} $$
The derivative of the function $\frac{1}{\sqrt{x}}$ with respect to $x$ is
$$ f'(x) = -\frac{1}{2\sqrt{x^3}} $$
More Information
Taking derivatives is a fundamental concept in calculus, and the example here shows how a function can be rewritten for simpler differentiation. The negative exponent signifies that the original function decreases as $x$ increases.
Tips
- Forgetting to apply the power rule correctly, leading to an incorrect derivative. To avoid this, remember to subtract 1 from the exponent after multiplying by the original exponent.
- Not rewriting the fraction in terms of exponents, which can complicate differentiation. Practice recognizing similar patterns for smoother calculations.
