What is the derivative of 1/5x?
Understand the Problem
The question is asking for the derivative of the function f(x) = 1/(5x), which is a mathematical problem about calculus. We will differentiate the function with respect to x.
Answer
The derivative is $f'(x) = -\frac{1}{5x^2}$.
Answer for screen readers
The derivative of the function is
$$ f'(x) = -\frac{1}{5x^2} $$
Steps to Solve
- Identify the function
The function we need to differentiate is given as:
$$ f(x) = \frac{1}{5x} $$
- Rewrite the function
To make differentiation easier, we can rewrite the function using negative exponents:
$$ f(x) = \frac{1}{5} \cdot x^{-1} $$
- Differentiate the function
Now we apply the power rule of differentiation, which states that if $f(x) = ax^n$, then $f'(x) = nax^{n-1}$.
Here, $a = \frac{1}{5}$ and $n = -1$. Thus,
$$ f'(x) = -1 \cdot \frac{1}{5} \cdot x^{-1-1} = -\frac{1}{5} x^{-2} $$
- Rewrite the derivative
We can rewrite the derivative back to fractional form:
$$ f'(x) = -\frac{1}{5x^2} $$
The derivative of the function is
$$ f'(x) = -\frac{1}{5x^2} $$
More Information
The derivative represents the rate of change of the function $f(x) = \frac{1}{5x}$ with respect to $x$. It indicates how steep the graph of the function is at any given point. The negative sign shows that the function is decreasing as $x$ increases.
Tips
- Forgetting to apply the power rule correctly.
- Not rewriting the function in a simpler form first.
- Misinterpreting the final answer and leaving it in negative exponent form instead of converting it back to fraction form.
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