What is the derivative of 2x with respect to x?
Understand the Problem
The question is asking for the derivative of the function 2x with respect to x. To solve this, we will apply the rules of differentiation, specifically the power rule.
Answer
The derivative of $f(x) = 2x$ is $2$.
Answer for screen readers
The derivative of the function $f(x) = 2x$ with respect to $x$ is $2$.
Steps to Solve
- Identify the function to differentiate
The function given is $f(x) = 2x$. We need to find the derivative of this function with respect to $x$.
- Apply the power rule
The power rule states that the derivative of $x^n$ is $nx^{n-1}$. Here, we can consider $f(x)$ as $f(x) = 2x^1$. We will differentiate using the power rule.
- Differentiate the function
According to the power rule, the derivative of $2x^1$ becomes: $$ f'(x) = 2 \cdot 1 \cdot x^{1-1} $$
- Simplify the derivative
Now we simplify the expression: $$ f'(x) = 2 \cdot 1 \cdot x^0 = 2 $$
Thus, the derivative of the function is simply $2$.
The derivative of the function $f(x) = 2x$ with respect to $x$ is $2$.
More Information
The derivative measures how the function $f(x)$ changes as $x$ changes. In this case, $f'(x) = 2$ means that for every unit increase in $x$, the function $f(x)$ increases by 2 units.
Tips
- Confusing the differentiation of a constant with that of a variable; remember that the derivative of any constant is always zero.
- Misapplying the power rule by failing to recognize the exponent of 1 in $2x$.
