Find the derivative of f(x) = 1/x^2
Understand the Problem
The question asks us to find the derivative of the function f(x) = 1/x^2. This requires applying the power rule after rewriting the function as f(x) = x^(-2). We will then simplify the result to express the derivative.
Answer
$f'(x) = \frac{-2}{x^3}$
Answer for screen readers
$f'(x) = \frac{-2}{x^3}$
Steps to Solve
- Rewrite the function using a negative exponent
Rewrite $f(x) = \frac{1}{x^2}$ as $f(x) = x^{-2}$ to prepare for the power rule.
- Apply the power rule
The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. Applying this rule to $f(x) = x^{-2}$, we get:
$f'(x) = -2x^{-2-1} = -2x^{-3}$
- Rewrite with a positive exponent
Rewrite the derivative with a positive exponent:
$f'(x) = -2x^{-3} = \frac{-2}{x^3}$
$f'(x) = \frac{-2}{x^3}$
More Information
The derivative of $f(x) = \frac{1}{x^2}$ is $f'(x) = \frac{-2}{x^3}$. This result tells us the slope of the tangent line to the curve of $f(x)$ at any point $x$.
Tips
A common mistake is forgetting to apply the chain rule when dealing with composite functions, but that's not an issue here. Another mistake is incorrectly applying the power rule by either adding instead of subtracting 1 from the exponent or making a sign error.
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