Calculus Examples

Find the Tangent Line at the Point e^(x^2+y^2)=xe^(5y)-y^2e^((5x)/2) , (2,1)
,
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
Differentiate using the chain rule, which states that is where and .
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Step 1.2.1.1
To apply the Chain Rule, set as .
Step 1.2.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.2.1.3
Replace all occurrences of with .
Step 1.2.2
Differentiate.
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Step 1.2.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Differentiate using the chain rule, which states that is where and .
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Step 1.2.3.1
To apply the Chain Rule, set as .
Step 1.2.3.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3.3
Replace all occurrences of with .
Step 1.2.4
Rewrite as .
Step 1.3
Differentiate the right side of the equation.
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Step 1.3.1
By the Sum Rule, the derivative of with respect to is .
Step 1.3.2
Evaluate .
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Step 1.3.2.1
Differentiate using the Product Rule which states that is where and .
Step 1.3.2.2
Differentiate using the chain rule, which states that is where and .
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Step 1.3.2.2.1
To apply the Chain Rule, set as .
Step 1.3.2.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.3.2.2.3
Replace all occurrences of with .
Step 1.3.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2.4
Rewrite as .
Step 1.3.2.5
Differentiate using the Power Rule which states that is where .
Step 1.3.2.6
Multiply by .
Step 1.3.3
Evaluate .
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Step 1.3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.3.2
Differentiate using the Product Rule which states that is where and .
Step 1.3.3.3
Differentiate using the chain rule, which states that is where and .
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Step 1.3.3.3.1
To apply the Chain Rule, set as .
Step 1.3.3.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.3.3.3.3
Replace all occurrences of with .
Step 1.3.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.3.5
Differentiate using the Power Rule which states that is where .
Step 1.3.3.6
Differentiate using the chain rule, which states that is where and .
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Step 1.3.3.6.1
To apply the Chain Rule, set as .
Step 1.3.3.6.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3.6.3
Replace all occurrences of with .
Step 1.3.3.7
Rewrite as .
Step 1.3.3.8
Multiply by .
Step 1.3.3.9
Combine and .
Step 1.3.3.10
Move to the left of .
Step 1.3.3.11
Combine and .
Step 1.3.3.12
Move to the left of .
Step 1.3.3.13
Move to the left of .
Step 1.3.4
Simplify.
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Step 1.3.4.1
Apply the distributive property.
Step 1.3.4.2
Multiply by .
Step 1.3.4.3
Reorder terms.
Step 1.3.4.4
Reorder factors in .
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
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 1.5.2
Reorder factors in .
Step 1.5.3
Simplify .
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Step 1.5.3.1
Apply the distributive property.
Step 1.5.3.2
Reorder.
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Step 1.5.3.2.1
Rewrite using the commutative property of multiplication.
Step 1.5.3.2.2
Rewrite using the commutative property of multiplication.
Step 1.5.3.2.3
Reorder factors in .
Step 1.5.4
Move all terms containing to the left side of the equation.
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Step 1.5.4.1
Subtract from both sides of the equation.
Step 1.5.4.2
To write as a fraction with a common denominator, multiply by .
Step 1.5.4.3
Combine and .
Step 1.5.4.4
Combine the numerators over the common denominator.
Step 1.5.4.5
Simplify the numerator.
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Step 1.5.4.5.1
Factor out of .
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Step 1.5.4.5.1.1
Factor out of .
Step 1.5.4.5.1.2
Factor out of .
Step 1.5.4.5.1.3
Factor out of .
Step 1.5.4.5.2
Multiply by .
Step 1.5.4.6
To write as a fraction with a common denominator, multiply by .
Step 1.5.4.7
Combine and .
Step 1.5.4.8
Combine the numerators over the common denominator.
Step 1.5.4.9
Simplify the numerator.
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Step 1.5.4.9.1
Multiply by .
Step 1.5.4.9.2
Apply the distributive property.
Step 1.5.4.9.3
Rewrite using the commutative property of multiplication.
Step 1.5.4.9.4
Rewrite using the commutative property of multiplication.
Step 1.5.4.9.5
Multiply by by adding the exponents.
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Step 1.5.4.9.5.1
Move .
Step 1.5.4.9.5.2
Multiply by .
Step 1.5.4.10
To write as a fraction with a common denominator, multiply by .
Step 1.5.4.11
Combine and .
Step 1.5.4.12
Combine the numerators over the common denominator.
Step 1.5.4.13
Multiply by .
Step 1.5.4.14
To write as a fraction with a common denominator, multiply by .
Step 1.5.4.15
Combine and .
Step 1.5.4.16
Combine the numerators over the common denominator.
Step 1.5.4.17
Move to the left of .
Step 1.5.4.18
Factor out of .
Step 1.5.4.19
Factor out of .
Step 1.5.4.20
Factor out of .
Step 1.5.4.21
Factor out of .
Step 1.5.4.22
Factor out of .
Step 1.5.4.23
Factor out of .
Step 1.5.4.24
Factor out of .
Step 1.5.4.25
Factor out of .
Step 1.5.4.26
Factor out of .
Step 1.5.4.27
Rewrite as .
Step 1.5.4.28
Move the negative in front of the fraction.
Step 1.5.4.29
Reorder factors in .
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
Dividing two negative values results in a positive value.
Step 1.5.5.2.2
Divide by .
Step 1.5.5.3
Simplify the right side.
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Step 1.5.5.3.1
Move the negative one from the denominator of .
Step 1.5.5.3.2
Rewrite as .
Step 1.5.5.3.3
Multiply by .
Step 1.5.6
Multiply both sides by .
Step 1.5.7
Simplify.
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Step 1.5.7.1
Simplify the left side.
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Step 1.5.7.1.1
Simplify .
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Step 1.5.7.1.1.1
Cancel the common factor of .
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Step 1.5.7.1.1.1.1
Cancel the common factor.
Step 1.5.7.1.1.1.2
Rewrite the expression.
Step 1.5.7.1.1.2
Reorder.
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Step 1.5.7.1.1.2.1
Move .
Step 1.5.7.1.1.2.2
Move .
Step 1.5.7.1.1.2.3
Move .
Step 1.5.7.1.1.2.4
Move .
Step 1.5.7.2
Simplify the right side.
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Step 1.5.7.2.1
Multiply by .
Step 1.5.8
Solve for .
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Step 1.5.8.1
Move all terms not containing to the right side of the equation.
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Step 1.5.8.1.1
Subtract from both sides of the equation.
Step 1.5.8.1.2
Add to both sides of the equation.
Step 1.5.8.2
Factor out of .
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Step 1.5.8.2.1
Factor out of .
Step 1.5.8.2.2
Factor out of .
Step 1.5.8.2.3
Factor out of .
Step 1.5.8.2.4
Factor out of .
Step 1.5.8.2.5
Factor out of .
Step 1.5.8.3
Divide each term in by and simplify.
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Step 1.5.8.3.1
Divide each term in by .
Step 1.5.8.3.2
Simplify the left side.
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Step 1.5.8.3.2.1
Cancel the common factor of and .
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Step 1.5.8.3.2.1.1
Factor out of .
Step 1.5.8.3.2.1.2
Cancel the common factors.
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Step 1.5.8.3.2.1.2.1
Factor out of .
Step 1.5.8.3.2.1.2.2
Factor out of .
Step 1.5.8.3.2.1.2.3
Factor out of .
Step 1.5.8.3.2.1.2.4
Factor out of .
Step 1.5.8.3.2.1.2.5
Factor out of .
Step 1.5.8.3.2.1.2.6
Cancel the common factor.
Step 1.5.8.3.2.1.2.7
Rewrite the expression.
Step 1.5.8.3.2.2
Cancel the common factor of .
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Step 1.5.8.3.2.2.1
Cancel the common factor.
Step 1.5.8.3.2.2.2
Divide by .
Step 1.5.8.3.3
Simplify the right side.
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Step 1.5.8.3.3.1
Simplify each term.
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Step 1.5.8.3.3.1.1
Cancel the common factor of and .
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Step 1.5.8.3.3.1.1.1
Factor out of .
Step 1.5.8.3.3.1.1.2
Cancel the common factors.
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Step 1.5.8.3.3.1.1.2.1
Factor out of .
Step 1.5.8.3.3.1.1.2.2
Factor out of .
Step 1.5.8.3.3.1.1.2.3
Factor out of .
Step 1.5.8.3.3.1.1.2.4
Factor out of .
Step 1.5.8.3.3.1.1.2.5
Factor out of .
Step 1.5.8.3.3.1.1.2.6
Cancel the common factor.
Step 1.5.8.3.3.1.1.2.7
Rewrite the expression.
Step 1.5.8.3.3.1.2
Move the negative in front of the fraction.
Step 1.5.8.3.3.1.3
Factor out of .
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Step 1.5.8.3.3.1.3.1
Factor out of .
Step 1.5.8.3.3.1.3.2
Factor out of .
Step 1.5.8.3.3.1.3.3
Factor out of .
Step 1.5.8.3.3.1.3.4
Factor out of .
Step 1.5.8.3.3.1.3.5
Factor out of .
Step 1.5.8.3.3.1.4
Move the negative in front of the fraction.
Step 1.5.8.3.3.1.5
Cancel the common factor of and .
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Step 1.5.8.3.3.1.5.1
Factor out of .
Step 1.5.8.3.3.1.5.2
Cancel the common factors.
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Step 1.5.8.3.3.1.5.2.1
Factor out of .
Step 1.5.8.3.3.1.5.2.2
Factor out of .
Step 1.5.8.3.3.1.5.2.3
Factor out of .
Step 1.5.8.3.3.1.5.2.4
Factor out of .
Step 1.5.8.3.3.1.5.2.5
Factor out of .
Step 1.5.8.3.3.1.5.2.6
Cancel the common factor.
Step 1.5.8.3.3.1.5.2.7
Rewrite the expression.
Step 1.5.8.3.3.2
To write as a fraction with a common denominator, multiply by .
Step 1.5.8.3.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.5.8.3.3.3.1
Multiply by .
Step 1.5.8.3.3.3.2
Reorder the factors of .
Step 1.5.8.3.3.4
Combine the numerators over the common denominator.
Step 1.5.8.3.3.5
Multiply by .
Step 1.5.8.3.3.6
To write as a fraction with a common denominator, multiply by .
Step 1.5.8.3.3.7
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.5.8.3.3.7.1
Multiply by .
Step 1.5.8.3.3.7.2
Reorder the factors of .
Step 1.5.8.3.3.8
Combine the numerators over the common denominator.
Step 1.5.8.3.3.9
Move to the left of .
Step 1.5.8.3.3.10
Simplify terms.
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Step 1.5.8.3.3.10.1
Factor out of .
Step 1.5.8.3.3.10.2
Factor out of .
Step 1.5.8.3.3.10.3
Factor out of .
Step 1.5.8.3.3.10.4
Factor out of .
Step 1.5.8.3.3.10.5
Factor out of .
Step 1.5.8.3.3.10.6
Rewrite as .
Step 1.5.8.3.3.10.7
Factor out of .
Step 1.5.8.3.3.10.8
Factor out of .
Step 1.5.8.3.3.10.9
Factor out of .
Step 1.5.8.3.3.10.10
Factor out of .
Step 1.5.8.3.3.10.11
Factor out of .
Step 1.5.8.3.3.10.12
Rewrite as .
Step 1.5.8.3.3.10.13
Cancel the common factor.
Step 1.5.8.3.3.10.14
Rewrite the expression.
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
Reduce the expression by cancelling the common factors.
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Step 1.7.3.1
Cancel the common factor of .
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Step 1.7.3.1.1
Cancel the common factor.
Step 1.7.3.1.2
Divide by .
Step 1.7.3.2
Cancel the common factor of .
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Step 1.7.3.2.1
Cancel the common factor.
Step 1.7.3.2.2
Divide by .
Step 1.7.4
Simplify the numerator.
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Step 1.7.4.1
Multiply by .
Step 1.7.4.2
Simplify each term.
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Step 1.7.4.2.1
Raise to the power of .
Step 1.7.4.2.2
One to any power is one.
Step 1.7.4.3
Add and .
Step 1.7.4.4
One to any power is one.
Step 1.7.4.5
Multiply by .
Step 1.7.4.6
Multiply by .
Step 1.7.4.7
Add and .
Step 1.7.4.8
Subtract from .
Step 1.7.5
Simplify the denominator.
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Step 1.7.5.1
Multiply by .
Step 1.7.5.2
Multiply by .
Step 1.7.5.3
Multiply by .
Step 1.7.5.4
Multiply by .
Step 1.7.5.5
Simplify each term.
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Step 1.7.5.5.1
Raise to the power of .
Step 1.7.5.5.2
One to any power is one.
Step 1.7.5.6
Add and .
Step 1.7.5.7
Subtract from .
Step 1.7.5.8
Subtract from .
Step 1.7.5.9
Multiply by .
Step 1.7.6
Cancel the common factor of .
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Step 1.7.6.1
Cancel the common factor.
Step 1.7.6.2
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
Cancel the common factor of .
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Step 2.3.1.5.1
Factor out of .
Step 2.3.1.5.2
Factor out of .
Step 2.3.1.5.3
Cancel the common factor.
Step 2.3.1.5.4
Rewrite the expression.
Step 2.3.1.6
Combine and .
Step 2.3.1.7
Simplify the expression.
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Step 2.3.1.7.1
Multiply by .
Step 2.3.1.7.2
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
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 2.3.2.5
Move the negative in front of the fraction.
Step 2.3.3
Reorder terms.
Step 3