What is the derivative of 1/3x?
Understand the Problem
The question is asking for the derivative of the function 1/3x, which involves applying the rules of differentiation to find the rate of change of the function with respect to x.
Answer
The derivative of the function is $f'(x) = \frac{1}{3}$.
Answer for screen readers
The derivative of the function $f(x) = \frac{1}{3}x$ is $f'(x) = \frac{1}{3}$.
Steps to Solve
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Identify the function to differentiate We are given the function $f(x) = \frac{1}{3}x$.
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Apply the power rule The power rule states that if $f(x) = ax^n$, then the derivative $f'(x) = nax^{n-1}$. Here, we can rewrite $f(x)$ as $\frac{1}{3} \cdot x^1$.
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Differentiate using the power rule Using the power rule, we find the derivative: $$ f'(x) = 1 \cdot \frac{1}{3} x^{1-1} = \frac{1}{3} x^0 = \frac{1}{3} $$
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Write the final answer The derivative of the function is simply the constant we calculated.
The derivative of the function $f(x) = \frac{1}{3}x$ is $f'(x) = \frac{1}{3}$.
More Information
The derivative represents the slope of the function at any point, and since our derivative is a constant, it indicates that the function has a constant rate of change.
Tips
- A common mistake is forgetting to apply the power rule correctly, especially with the exponent.
- Another mistake might be neglecting to simplify the expression appropriately.
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