What is the differentiation of ax?
Understand the Problem
The question is asking for the derivative of the function ax, which is a basic differentiation problem in calculus. The approach to solve this involves applying the power rule of differentiation.
Answer
The derivative is $a$.
Answer for screen readers
The derivative of the function $f(x) = ax$ is $a$.
Steps to Solve
- Identify the function to differentiate
The function we want to differentiate is $f(x) = ax$, where $a$ is a constant.
- Apply the power rule
The power rule states that if $f(x) = x^n$, then the derivative $f'(x) = n \cdot x^{n-1}$. In our case, we can rewrite $f(x)$ as $f(x) = a \cdot x^1$.
- Differentiate using the power rule
By applying the power rule, the derivative of $f(x) = a \cdot x^1$ becomes:
$$ f'(x) = 1 \cdot a \cdot x^{1-1} = a \cdot x^0 = a $$
- Conclude the result
The derivative of the function $f(x) = ax$ is simply $a$.
The derivative of the function $f(x) = ax$ is $a$.
More Information
This result illustrates a fundamental concept in calculus. The derivative of a linear function with a constant coefficient is simply that constant. This principle is widely applied in various fields, such as physics, engineering, and economics.
Tips
- Confusing coefficients with variables: Remember that 'a' is a constant, so the derivative simplifies directly.
- Misapplying the power rule: Ensure the format of the function aligns with the conditions of the power rule.
