Calculus Examples

Find the Tangent Line at the Point x^2+xy-y^2=-1 , (1,2)
,
Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
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Step 1.1
Differentiate both sides of the equation.
Step 1.2
Differentiate the left side of the equation.
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Step 1.2.1
Differentiate.
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Step 1.2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2
Evaluate .
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Step 1.2.2.1
Differentiate using the Product Rule which states that is where and .
Step 1.2.2.2
Rewrite as .
Step 1.2.2.3
Differentiate using the Power Rule which states that is where .
Step 1.2.2.4
Multiply by .
Step 1.2.3
Evaluate .
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Step 1.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.3.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.3.2.1
To apply the Chain Rule, set as .
Step 1.2.3.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3.2.3
Replace all occurrences of with .
Step 1.2.3.3
Rewrite as .
Step 1.2.3.4
Multiply by .
Step 1.2.4
Reorder terms.
Step 1.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.4
Reform the equation by setting the left side equal to the right side.
Step 1.5
Solve for .
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Step 1.5.1
Move all terms not containing to the right side of the equation.
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Step 1.5.1.1
Subtract from both sides of the equation.
Step 1.5.1.2
Subtract from both sides of the equation.
Step 1.5.2
Factor out of .
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Step 1.5.2.1
Factor out of .
Step 1.5.2.2
Factor out of .
Step 1.5.2.3
Factor out of .
Step 1.5.3
Divide each term in by and simplify.
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Step 1.5.3.1
Divide each term in by .
Step 1.5.3.2
Simplify the left side.
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Step 1.5.3.2.1
Cancel the common factor of .
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Step 1.5.3.2.1.1
Cancel the common factor.
Step 1.5.3.2.1.2
Divide by .
Step 1.5.3.3
Simplify the right side.
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Step 1.5.3.3.1
Combine the numerators over the common denominator.
Step 1.5.3.3.2
Factor out of .
Step 1.5.3.3.3
Factor out of .
Step 1.5.3.3.4
Factor out of .
Step 1.5.3.3.5
Simplify the expression.
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Step 1.5.3.3.5.1
Rewrite as .
Step 1.5.3.3.5.2
Move the negative in front of the fraction.
Step 1.6
Replace with .
Step 1.7
Evaluate at and .
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Step 1.7.1
Replace the variable with in the expression.
Step 1.7.2
Replace the variable with in the expression.
Step 1.7.3
Remove parentheses.
Step 1.7.4
Simplify the numerator.
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Step 1.7.4.1
Multiply by .
Step 1.7.4.2
Add and .
Step 1.7.5
Simplify the denominator.
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Step 1.7.5.1
Multiply by .
Step 1.7.5.2
Subtract from .
Step 1.7.6
Move the negative in front of the fraction.
Step 1.7.7
Multiply .
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Step 1.7.7.1
Multiply by .
Step 1.7.7.2
Multiply by .
Step 2
Plug the slope and point values into the point-slope formula and solve for .
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Step 2.1
Use the slope and a given point to substitute for and in the point-slope form , which is derived from the slope equation .
Step 2.2
Simplify the equation and keep it in point-slope form.
Step 2.3
Solve for .
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Step 2.3.1
Simplify .
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Step 2.3.1.1
Rewrite.
Step 2.3.1.2
Simplify by adding zeros.
Step 2.3.1.3
Apply the distributive property.
Step 2.3.1.4
Combine and .
Step 2.3.1.5
Multiply .
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Step 2.3.1.5.1
Combine and .
Step 2.3.1.5.2
Multiply by .
Step 2.3.1.6
Move the negative in front of the fraction.
Step 2.3.2
Move all terms not containing to the right side of the equation.
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Step 2.3.2.1
Add to both sides of the equation.
Step 2.3.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.3.2.3
Combine and .
Step 2.3.2.4
Combine the numerators over the common denominator.
Step 2.3.2.5
Simplify the numerator.
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Step 2.3.2.5.1
Multiply by .
Step 2.3.2.5.2
Add and .
Step 2.3.3
Reorder terms.
Step 3