What is the derivative of 5x?
Understand the Problem
The question is asking for the derivative of the function 5x with respect to x. To solve this, we will apply the rules of differentiation.
Answer
$5$
Answer for screen readers
The derivative of the function $5x$ with respect to $x$ is $5$.
Steps to Solve
- Identify the function to differentiate
The function we are differentiating is $f(x) = 5x$.
- Apply the power rule of differentiation
According to the power rule, if we have a term in the form of $ax^n$, the derivative is given by $n \cdot ax^{n-1}$.
For our function:
- Here, $a = 5$ and $n = 1$.
- Therefore, the derivative is calculated as follows:
$$ f'(x) = 1 \cdot 5x^{1-1} $$
- Simplify the derivative
Now we simplify the expression obtained from the previous step:
$$ f'(x) = 5x^{0} $$
Since any number raised to the power of 0 is 1, we have:
$$ f'(x) = 5 \cdot 1 = 5 $$
The derivative of the function $5x$ with respect to $x$ is $5$.
More Information
The derivative tells us the rate of change of the function at any point. In this case, the constant derivative of $5$ indicates that the function is increasing at a constant rate of $5$ units for every 1 unit increase in $x$.
Tips
- Forgetting the power rule: It's essential to remember to apply the power rule correctly. In this case, students may overlook the fact that $n = 1$.
- Incorrectly simplifying $x^0$: Remember that $x^0 = 1$, which is an important step in simplifying the derivative.
