text: # Partial Derivative Calculator
  
  ## Calculate partial derivatives step by step
  
  This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration.
  
  Enter a function:
  
  Enter the order of integration:
  
  Hint: type x^2,y to calculate (partial^3 f)/(partial x^2 partial y), or enter x,y^2,x to find (partial^4 f)/(partial x partial y^2 partial x).
  
  If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.
  
  ### Solution
  
  Your input: find $\frac{\partial}{\partial y}\left(3 x + 4 y\right)$
  
  The derivative of a sum/difference is the sum/difference of derivatives:
  
  $${\color{red}{\frac{\partial}{\partial y}\left(3 x + 4 y\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(3 x\right) + \frac{\partial}{\partial y}\left(4 y\right)\right)}}$$
  
  Apply the constant multiple rule $\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$ with $c=4$ and $f=y$:
  
  $${\color{red}{\frac{\partial}{\partial y}\left(4 y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)={\color{red}{\left(4 \frac{\partial}{\partial y}\left(y\right)\right)}} + \frac{\partial}{\partial y}\left(3 x\right)$$
  
  Apply the power rule $\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$ with $n=1$, in other words $\frac{\partial}{\partial y} \left(y \right)=1$:
  
  $$4 {\color{red}{\frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)=4 {\color{red}{1}} + \frac{\partial}{\partial y}\left(3 x\right)$$
  
  The derivative of a constant is 0:
  
  $$4 + {\color{red}{\frac{\partial}{\partial y}\left(3 x\right)}}=4 + {\color{red}{\left(0\right)}}$$
  
  Thus, $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$
  
  Answer: $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$
explanation: The document is a step-by-step solution for calculating a partial derivative. It includes a specific question about finding the partial derivative of a function and provides a detailed solution to that question.
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: # Partial Derivative Calculator
  
  ## Calculate partial derivatives step by step
  
  This online calculator will calculate the partial derivative of the function, with steps shown. You can specify any order of integration.
  
  Enter a function:
  
  Enter the order of integration:
  
  Hint: type x^2,y to calculate (partial^3 f)/(partial x^2 partial y), or enter x,y^2,x to find (partial^4 f)/(partial x partial y^2 partial x).
  
  If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.
  
  ### Solution
  
  Your input: find $\frac{\partial}{\partial y}\left(3 x + 4 y\right)$
  
  The derivative of a sum/difference is the sum/difference of derivatives:
  
  $${\color{red}{\frac{\partial}{\partial y}\left(3 x + 4 y\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(3 x\right) + \frac{\partial}{\partial y}\left(4 y\right)\right)}}$$
  
  Apply the constant multiple rule $\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$ with $c=4$ and $f=y$:
  
  $${\color{red}{\frac{\partial}{\partial y}\left(4 y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)={\color{red}{\left(4 \frac{\partial}{\partial y}\left(y\right)\right)}} + \frac{\partial}{\partial y}\left(3 x\right)$$
  
  Apply the power rule $\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$ with $n=1$, in other words $\frac{\partial}{\partial y} \left(y \right)=1$:
  
  $$4 {\color{red}{\frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(3 x\right)=4 {\color{red}{1}} + \frac{\partial}{\partial y}\left(3 x\right)$$
  
  The derivative of a constant is 0:
  
  $$4 + {\color{red}{\frac{\partial}{\partial y}\left(3 x\right)}}=4 + {\color{red}{\left(0\right)}}$$
  
  Thus, $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$
  
  Answer: $\frac{\partial}{\partial y}\left(3 x + 4 y\right)=4$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a step-by-step solution for calculating a partial derivative. It includes a specific question about finding the partial derivative of a function and provides a detailed solution to that question.
  
  Label: question-and-answer