What is the derivative of log10(x)?
Understand the Problem
The question is asking for the derivative of the function log10(x), which is a logarithmic function. To solve this, we will apply the rules of differentiation for logarithmic functions.
Answer
The derivative of $\log_{10}(x)$ is $f'(x) = \frac{1}{x \ln(10)}$.
Answer for screen readers
The derivative of the function $\log_{10}(x)$ is
$$ f'(x) = \frac{1}{x \ln(10)} $$
Steps to Solve
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Identify the function and the differentiation rule
We need to differentiate the function $f(x) = \log_{10}(x)$. The differentiation rule for logarithmic functions states that the derivative of $\log_b(x)$ is given by:
$$ \frac{d}{dx}\log_b(x) = \frac{1}{x \ln(b)} $$
where $b$ is the base of the logarithm and $\ln$ is the natural logarithm.
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Apply the differentiation rule
For our function, the base $b$ is 10. Plugging that into the formula, we get:
$$ \frac{d}{dx}\log_{10}(x) = \frac{1}{x \ln(10)} $$
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Simplify the result if necessary
The derivative of the function can be expressed as:
$$ f'(x) = \frac{1}{x \ln(10)} $$
This is already in its simplest form.
The derivative of the function $\log_{10}(x)$ is
$$ f'(x) = \frac{1}{x \ln(10)} $$
More Information
The derivative $\frac{1}{x \ln(10)}$ indicates how the function $\log_{10}(x)$ changes with respect to $x$. The logarithm's base, in this case, 10, is important in scaling the derivative.
Tips
- Ignoring the base when differentiating: Always remember to include the logarithm base in the derivative formula.
- Not simplifying correctly: Make sure to express the final result in the simplest form.
