# 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

$\frac{d y}{d x} = \frac{d y}{d t} \cdot \frac{d t}{d x} = \frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}$

$d y = \left(\frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}\right) \cdot d x$
The passage contains a question about differentiating a parametric equation and provides a detailed step-by-step solution to the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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

$\frac{d y}{d x} = \frac{d y}{d t} \cdot \frac{d t}{d x} = \frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}$

$d y = \left(\frac{2 \left(t - 2\right) + 1}{2 \left(t + 1\right) + {e}^{t} \cdot l n e}\right) \cdot d x$
user
The passage contains a question about differentiating a parametric equation and provides a detailed step-by-step solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER