# Calculus Examples

Find the Tangent Line at the Point 2x^2+y^2=12 , (2,-2)
,
Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
Differentiate both sides of the equation.
Differentiate the left side of the equation.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Rewrite as .
Reorder terms.
Since is constant with respect to , the derivative of with respect to is .
Reform the equation by setting the left side equal to the right side.
Solve for .
Subtract from both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Replace with .
Evaluate at and .
Replace the variable with in the expression.
Replace the variable with in the expression.
Move the negative one from the denominator of .
Multiply .
Multiply by .
Multiply by .
Step 2
Plug the slope and point values into the point-slope formula and solve for .
Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .
Simplify the equation and keep it in point-slope form.
Solve for .
Simplify .
Rewrite.
Simplify by adding zeros.
Apply the distributive property.
Multiply by .
Move all terms not containing to the right side of the equation.
Subtract from both sides of the equation.
Subtract from .
Step 3
The text passage provides a detailed step-by-step solution to finding the tangent line to the curve \(2x^2 + y^2 = 12\) at the point \((2, -2)\).
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
# Calculus Examples

Find the Tangent Line at the Point 2x^2+y^2=12 , (2,-2)
,
Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
Differentiate both sides of the equation.
Differentiate the left side of the equation.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Rewrite as .
Reorder terms.
Since is constant with respect to , the derivative of with respect to is .
Reform the equation by setting the left side equal to the right side.
Solve for .
Subtract from both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Replace with .
Evaluate at and .
Replace the variable with in the expression.
Replace the variable with in the expression.
Move the negative one from the denominator of .
Multiply .
Multiply by .
Multiply by .
Step 2
Plug the slope and point values into the point-slope formula and solve for .
Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .
Simplify the equation and keep it in point-slope form.
Solve for .
Simplify .
Rewrite.
Simplify by adding zeros.
Apply the distributive property.
Multiply by .
Move all terms not containing to the right side of the equation.
Subtract from both sides of the equation.
Subtract from .
Step 3
user
The text passage provides a detailed step-by-step solution to finding the tangent line to the curve \(2x^2 + y^2 = 12\) at the point \((2, -2)\).

TYPE: SINGLE_QUESTION_WITH_ANSWER