Find the derivative of f(x)= 1/x
Understand the Problem
The question asks to find the derivative of the function f(x) = 1/x. This involves applying the power rule or quotient rule of differentiation.
Answer
$f'(x) = -\frac{1}{x^2}$
Answer for screen readers
$f'(x) = -\frac{1}{x^2}$
Steps to Solve
- Rewrite the function using a negative exponent
Rewrite $f(x) = \frac{1}{x}$ as $f(x) = x^{-1}$. This makes it easier to apply 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 to our function:
$f'(x) = (-1)x^{-1-1}$
- Simplify the result
Simplify the exponent and rewrite the expression:
$f'(x) = -x^{-2}$
$f'(x) = -\frac{1}{x^2}$
$f'(x) = -\frac{1}{x^2}$
More Information
The derivative of $f(x) = \frac{1}{x}$ is $f'(x) = -\frac{1}{x^2}$. This indicates that the slope of the tangent line to the curve $f(x) = \frac{1}{x}$ is always negative for any $x \neq 0$.
Tips
A common mistake is incorrectly applying the power rule, especially with negative exponents. For example, forgetting to subtract 1 from the exponent or making a sign error. Another mistake could be not simplifying the final answer.
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