🎧 New: AI-Generated Podcasts Turn your study notes into engaging audio conversations. Learn more

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

  1. Identify the function

The function we need to differentiate is given as:

$$ f(x) = \frac{1}{5x} $$

  1. Rewrite the function

To make differentiation easier, we can rewrite the function using negative exponents:

$$ f(x) = \frac{1}{5} \cdot x^{-1} $$

  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,

$$ f'(x) = -1 \cdot \frac{1}{5} \cdot x^{-1-1} = -\frac{1}{5} x^{-2} $$

  1. 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.
Thank you for voting!
Use Quizgecko on...
Browser
Browser