Calculus Examples

Find the Tangent Line at the Point 2(x^2+y^2)^2=25(x^2-y^2) , (3,1)
,
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
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.2.1
To apply the Chain Rule, set as .
Step 1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2.3
Replace all occurrences of with .
Step 1.2.3
Differentiate.
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Step 1.2.3.1
Multiply by .
Step 1.2.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.2.3.3
Differentiate using the Power Rule which states that is where .
Step 1.2.4
Differentiate using the chain rule, which states that is where and .
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Step 1.2.4.1
To apply the Chain Rule, set as .
Step 1.2.4.2
Differentiate using the Power Rule which states that is where .
Step 1.2.4.3
Replace all occurrences of with .
Step 1.2.5
Rewrite as .
Step 1.2.6
Simplify.
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Step 1.2.6.1
Apply the distributive property.
Step 1.2.6.2
Reorder the factors of .
Step 1.3
Differentiate the right side of the equation.
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Step 1.3.1
Differentiate.
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Step 1.3.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.1.2
By the Sum Rule, the derivative of with respect to is .
Step 1.3.1.3
Differentiate using the Power Rule which states that is where .
Step 1.3.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the chain rule, which states that is where and .
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Step 1.3.2.1
To apply the Chain Rule, set as .
Step 1.3.2.2
Differentiate using the Power Rule which states that is where .
Step 1.3.2.3
Replace all occurrences of with .
Step 1.3.3
Multiply by .
Step 1.3.4
Rewrite as .
Step 1.3.5
Simplify.
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Step 1.3.5.1
Apply the distributive property.
Step 1.3.5.2
Combine terms.
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Step 1.3.5.2.1
Multiply by .
Step 1.3.5.2.2
Multiply by .
Step 1.3.5.3
Reorder terms.
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
Simplify .
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Step 1.5.1.1
Rewrite.
Step 1.5.1.2
Simplify by adding zeros.
Step 1.5.1.3
Expand using the FOIL Method.
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Step 1.5.1.3.1
Apply the distributive property.
Step 1.5.1.3.2
Apply the distributive property.
Step 1.5.1.3.3
Apply the distributive property.
Step 1.5.1.4
Simplify each term.
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Step 1.5.1.4.1
Rewrite using the commutative property of multiplication.
Step 1.5.1.4.2
Multiply by by adding the exponents.
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Step 1.5.1.4.2.1
Move .
Step 1.5.1.4.2.2
Multiply by .
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Step 1.5.1.4.2.2.1
Raise to the power of .
Step 1.5.1.4.2.2.2
Use the power rule to combine exponents.
Step 1.5.1.4.2.3
Add and .
Step 1.5.1.4.3
Multiply by .
Step 1.5.1.4.4
Rewrite using the commutative property of multiplication.
Step 1.5.1.4.5
Multiply by .
Step 1.5.1.4.6
Multiply by .
Step 1.5.1.4.7
Multiply by by adding the exponents.
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Step 1.5.1.4.7.1
Move .
Step 1.5.1.4.7.2
Multiply by .
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Step 1.5.1.4.7.2.1
Raise to the power of .
Step 1.5.1.4.7.2.2
Use the power rule to combine exponents.
Step 1.5.1.4.7.3
Add and .
Step 1.5.1.4.8
Multiply by .
Step 1.5.2
Add to both sides of the equation.
Step 1.5.3
Move all terms not containing to the right side of the equation.
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Step 1.5.3.1
Subtract from both sides of the equation.
Step 1.5.3.2
Subtract from both sides of the equation.
Step 1.5.4
Factor out of .
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Step 1.5.4.1
Factor out of .
Step 1.5.4.2
Factor out of .
Step 1.5.4.3
Factor out of .
Step 1.5.4.4
Factor out of .
Step 1.5.4.5
Factor out of .
Step 1.5.5
Divide each term in by and simplify.
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Step 1.5.5.1
Divide each term in by .
Step 1.5.5.2
Simplify the left side.
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Step 1.5.5.2.1
Cancel the common factor of .
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Step 1.5.5.2.1.1
Cancel the common factor.
Step 1.5.5.2.1.2
Rewrite the expression.
Step 1.5.5.2.2
Cancel the common factor of .
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Step 1.5.5.2.2.1
Cancel the common factor.
Step 1.5.5.2.2.2
Rewrite the expression.
Step 1.5.5.2.3
Cancel the common factor of .
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Step 1.5.5.2.3.1
Cancel the common factor.
Step 1.5.5.2.3.2
Divide by .
Step 1.5.5.3
Simplify the right side.
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Step 1.5.5.3.1
Simplify each term.
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Step 1.5.5.3.1.1
Cancel the common factor of and .
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Step 1.5.5.3.1.1.1
Factor out of .
Step 1.5.5.3.1.1.2
Cancel the common factors.
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Step 1.5.5.3.1.1.2.1
Factor out of .
Step 1.5.5.3.1.1.2.2
Cancel the common factor.
Step 1.5.5.3.1.1.2.3
Rewrite the expression.
Step 1.5.5.3.1.2
Cancel the common factor of and .
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Step 1.5.5.3.1.2.1
Factor out of .
Step 1.5.5.3.1.2.2
Cancel the common factors.
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Step 1.5.5.3.1.2.2.1
Factor out of .
Step 1.5.5.3.1.2.2.2
Cancel the common factor.
Step 1.5.5.3.1.2.2.3
Rewrite the expression.
Step 1.5.5.3.1.3
Move the negative in front of the fraction.
Step 1.5.5.3.1.4
Cancel the common factor of and .
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Step 1.5.5.3.1.4.1
Factor out of .
Step 1.5.5.3.1.4.2
Cancel the common factors.
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Step 1.5.5.3.1.4.2.1
Factor out of .
Step 1.5.5.3.1.4.2.2
Cancel the common factor.
Step 1.5.5.3.1.4.2.3
Rewrite the expression.
Step 1.5.5.3.1.5
Cancel the common factor of and .
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Step 1.5.5.3.1.5.1
Factor out of .
Step 1.5.5.3.1.5.2
Cancel the common factors.
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Step 1.5.5.3.1.5.2.1
Cancel the common factor.
Step 1.5.5.3.1.5.2.2
Rewrite the expression.
Step 1.5.5.3.1.6
Move the negative in front of the fraction.
Step 1.5.5.3.2
To write as a fraction with a common denominator, multiply by .
Step 1.5.5.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.5.5.3.3.1
Multiply by .
Step 1.5.5.3.3.2
Reorder the factors of .
Step 1.5.5.3.4
Combine the numerators over the common denominator.
Step 1.5.5.3.5
Combine the numerators over the common denominator.
Step 1.5.5.3.6
Multiply by by adding the exponents.
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Step 1.5.5.3.6.1
Move .
Step 1.5.5.3.6.2
Multiply by .
Step 1.5.5.3.7
Factor out of .
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Step 1.5.5.3.7.1
Factor out of .
Step 1.5.5.3.7.2
Factor out of .
Step 1.5.5.3.7.3
Factor out of .
Step 1.5.5.3.7.4
Factor out of .
Step 1.5.5.3.7.5
Factor out of .
Step 1.5.5.3.8
Factor out of .
Step 1.5.5.3.9
Rewrite as .
Step 1.5.5.3.10
Factor out of .
Step 1.5.5.3.11
Factor out of .
Step 1.5.5.3.12
Factor out of .
Step 1.5.5.3.13
Simplify the expression.
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Step 1.5.5.3.13.1
Rewrite as .
Step 1.5.5.3.13.2
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
Simplify the numerator.
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Step 1.7.3.1
Raise to the power of .
Step 1.7.3.2
Multiply by .
Step 1.7.3.3
One to any power is one.
Step 1.7.3.4
Multiply by .
Step 1.7.3.5
Subtract from .
Step 1.7.3.6
Add and .
Step 1.7.4
Multiply.
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Step 1.7.4.1
Multiply by .
Step 1.7.4.2
Multiply by .
Step 1.7.5
Simplify the denominator.
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Step 1.7.5.1
Raise to the power of .
Step 1.7.5.2
Multiply by .
Step 1.7.5.3
One to any power is one.
Step 1.7.5.4
Multiply by .
Step 1.7.5.5
Add and .
Step 1.7.5.6
Add and .
Step 1.7.6
Cancel the common factor of and .
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Step 1.7.6.1
Factor out of .
Step 1.7.6.2
Cancel the common factors.
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Step 1.7.6.2.1
Factor out of .
Step 1.7.6.2.2
Cancel the common factor.
Step 1.7.6.2.3
Rewrite the expression.
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
Multiply by .
Step 2.3.1.5.2
Combine and .
Step 2.3.1.5.3
Multiply by .
Step 2.3.1.6
Move to the left of .
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
Write as a fraction with a common denominator.
Step 2.3.2.3
Combine the numerators over the common denominator.
Step 2.3.2.4
Add and .
Step 2.3.3
Write in form.
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Step 2.3.3.1
Reorder terms.
Step 2.3.3.2
Remove parentheses.
Step 3