text: # How do you differentiate the following parametric equation:  x(t)=(t+1)^2+e^t, y(t)= (t-2)^2+t?
  
  May 27, 2016
  
  $d y = \left(\frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}\right) \cdot d x$
  
  #### Explanation:
  
  $\frac{d x}{d t} = 2 \cdot \left(t + 1\right) \cdot 1 + {e}^{t} \cdot l n e = 2 \left(t + 1\right) + {e}^{t} \cdot l n e$
  
  $\frac{d y}{d t} = 2 \cdot \left(t - 2\right) \cdot 1 + 1 =$2(t-2)+1
  
  $\frac{d y}{d x} = \frac{d y}{d t} \cdot \frac{d t}{d x} = \frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}$
  
  $d y = \left(\frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}\right) \cdot d x$
explanation: The document contains a math question about differentiating a parametric equation, followed by a detailed 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: # How do you differentiate the following parametric equation:  x(t)=(t+1)^2+e^t, y(t)= (t-2)^2+t?
  
  May 27, 2016
  
  $d y = \left(\frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}\right) \cdot d x$
  
  #### Explanation:
  
  $\frac{d x}{d t} = 2 \cdot \left(t + 1\right) \cdot 1 + {e}^{t} \cdot l n e = 2 \left(t + 1\right) + {e}^{t} \cdot l n e$
  
  $\frac{d y}{d t} = 2 \cdot \left(t - 2\right) \cdot 1 + 1 =$2(t-2)+1
  
  $\frac{d y}{d x} = \frac{d y}{d t} \cdot \frac{d t}{d x} = \frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}$
  
  $d y = \left(\frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}\right) \cdot d x$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question about differentiating a parametric equation, followed by a detailed step-by-step solution to the problem.
  
  Label: question-and-answer