Calculus Examples

Find the Tangent Line at the Point 2x^2+y^2=12 , (2,-2)
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Step 1
Find the first derivative and evaluate at and to find the slope of the tangent line.
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Differentiate both sides of the equation.
Differentiate the left side of the equation.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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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 .
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Differentiate using the chain rule, which states that is where and .
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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 .
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Subtract from both sides of the equation.
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify the right side.
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Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Replace with .
Evaluate at and .
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Replace the variable with in the expression.
Replace the variable with in the expression.
Move the negative one from the denominator of .
Multiply .
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Multiply by .
Multiply by .
Step 2
Plug the slope and point values into the point-slope formula and solve for .
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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 .
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Simplify .
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Rewrite.
Simplify by adding zeros.
Apply the distributive property.
Multiply by .
Move all terms not containing to the right side of the equation.
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Subtract from both sides of the equation.
Subtract from .
Step 3
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