text: # When taking the derivative of $8 \log \frac{x}{10}$, do we ignore the factor of $\frac{1}{10}$? Why?
  
  In the midterm of probability in my university, they demand for us to calculate the derivative of $$h(x)=8\times\log_e{(\frac{x}{10})}$$ and i find $$h(x)^=\frac{8}{10x}$$ but in the solution they write $$h(x)^=\frac{8}{x}$$ , i repeat the operation a lot of time but i can't find the same result, i know that this derivative was simple but i was really confused! how to find the same result and why we ignore 10 ? the second question, in the exponential low, we find $$\lambda = \frac{1}{8}$$ and in the correction of the teacher he write that the $$E(x)=\frac{1}{8}$$ not 8, i really want to know how!
  
  • 1) How do you find $\frac8{10x}$? A still wrong but less surprising answer would be $\frac8{x/10}$. 2) Hint: $\log(x/10)=\log(x)-\log(10)$, and the derivative of a constant is $0$. Apr 21 at 6:40
  
  Looks like you forgot the chain rule: $$\frac{d}{dx}8\ln\left(\frac{x}{10}\right)=\frac{8}{x/10}\cdot\frac{d}{dx}\frac{x}{10}=\frac{10\cdot8}{x}\cdot\frac{1}{10}=\frac{8}{x}.$$
  Since $$\ln\left(\frac x{10}\right)=\ln(x)-\ln(10)$$, and $$\ln(10)$$ is constant, the derivative of $$\ln\left(\frac x{10}\right)$$ is the same as that of $$\ln(x)$$.
  In general, if $$f(x)=\ln(ax)$$, then $$f'(x)=\frac1x$$, and we can ignore the constant $$a$$ inside the logarithm. Geometrically this is because for the logarithm, stretching or compressing the graph in the $$x$$-direction is equivalent to moving it in the $$y$$-direction. (Which is related to the fact that moving the graph of the exponential function in the $$x$$-direction is equivalent to stretching it in the $$y$$-direction due to $$e^{x-a}=e^{-a}e^x$$.)
explanation: The text extract contains a math question and an answer to the question. The user asks about the derivative of a logarithmic function and why a specific factor is ignored. The response provides a detailed explanation and step-by-step solution to the problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # When taking the derivative of $8 \log \frac{x}{10}$, do we ignore the factor of $\frac{1}{10}$? Why?
  
  In the midterm of probability in my university, they demand for us to calculate the derivative of $$h(x)=8\times\log_e{(\frac{x}{10})}$$ and i find $$h(x)^=\frac{8}{10x}$$ but in the solution they write $$h(x)^=\frac{8}{x}$$ , i repeat the operation a lot of time but i can't find the same result, i know that this derivative was simple but i was really confused! how to find the same result and why we ignore 10 ? the second question, in the exponential low, we find $$\lambda = \frac{1}{8}$$ and in the correction of the teacher he write that the $$E(x)=\frac{1}{8}$$ not 8, i really want to know how!
  
  • 1) How do you find $\frac8{10x}$? A still wrong but less surprising answer would be $\frac8{x/10}$. 2) Hint: $\log(x/10)=\log(x)-\log(10)$, and the derivative of a constant is $0$. Apr 21 at 6:40
  
  Looks like you forgot the chain rule: $$\frac{d}{dx}8\ln\left(\frac{x}{10}\right)=\frac{8}{x/10}\cdot\frac{d}{dx}\frac{x}{10}=\frac{10\cdot8}{x}\cdot\frac{1}{10}=\frac{8}{x}.$$
  Since $$\ln\left(\frac x{10}\right)=\ln(x)-\ln(10)$$, and $$\ln(10)$$ is constant, the derivative of $$\ln\left(\frac x{10}\right)$$ is the same as that of $$\ln(x)$$.
  In general, if $$f(x)=\ln(ax)$$, then $$f'(x)=\frac1x$$, and we can ignore the constant $$a$$ inside the logarithm. Geometrically this is because for the logarithm, stretching or compressing the graph in the $$x$$-direction is equivalent to moving it in the $$y$$-direction. (Which is related to the fact that moving the graph of the exponential function in the $$x$$-direction is equivalent to stretching it in the $$y$$-direction due to $$e^{x-a}=e^{-a}e^x$$.)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math question and an answer to the question. The user asks about the derivative of a logarithmic function and why a specific factor is ignored. The response provides a detailed explanation and step-by-step solution to the problem.
  
  Label: question-and-answer