Calculus Examples

Find dy/dx x/y+ square root of x+y=e^(xy)
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
Differentiate the left side of the equation.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
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Step 3.2.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Rewrite as .
Step 3.2.4
Multiply by .
Step 3.3
Evaluate .
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Step 3.3.1
Differentiate using the chain rule, which states that is where and .
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Step 3.3.1.1
To apply the Chain Rule, set as .
Step 3.3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.3.1.3
Replace all occurrences of with .
Step 3.3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3.3
Differentiate using the Power Rule which states that is where .
Step 3.3.4
Rewrite as .
Step 3.3.5
To write as a fraction with a common denominator, multiply by .
Step 3.3.6
Combine and .
Step 3.3.7
Combine the numerators over the common denominator.
Step 3.3.8
Simplify the numerator.
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Step 3.3.8.1
Multiply by .
Step 3.3.8.2
Subtract from .
Step 3.3.9
Move the negative in front of the fraction.
Step 3.3.10
Combine and .
Step 3.3.11
Move to the denominator using the negative exponent rule .
Step 3.4
Simplify.
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Step 3.4.1
Combine terms.
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Step 3.4.1.1
To write as a fraction with a common denominator, multiply by .
Step 3.4.1.2
Combine and .
Step 3.4.1.3
Combine the numerators over the common denominator.
Step 3.4.1.4
Combine and .
Step 3.4.2
Reorder terms.
Step 4
Differentiate the right side of the equation.
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Step 4.1
Differentiate using the chain rule, which states that is where and .
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Step 4.1.1
To apply the Chain Rule, set as .
Step 4.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.1.3
Replace all occurrences of with .
Step 4.2
Differentiate using the Product Rule which states that is where and .
Step 4.3
Rewrite as .
Step 4.4
Differentiate using the Power Rule which states that is where .
Step 4.5
Multiply by .
Step 4.6
Simplify.
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Step 4.6.1
Apply the distributive property.
Step 4.6.2
Reorder terms.
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Solve for .
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Step 6.1
Multiply both sides by .
Step 6.2
Simplify.
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Step 6.2.1
Simplify the left side.
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Step 6.2.1.1
Simplify .
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Step 6.2.1.1.1
Cancel the common factor of .
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Step 6.2.1.1.1.1
Cancel the common factor.
Step 6.2.1.1.1.2
Rewrite the expression.
Step 6.2.1.1.2
Multiply by .
Step 6.2.1.1.3
To write as a fraction with a common denominator, multiply by .
Step 6.2.1.1.4
Simplify terms.
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Step 6.2.1.1.4.1
Combine and .
Step 6.2.1.1.4.2
Combine the numerators over the common denominator.
Step 6.2.1.1.5
Simplify the numerator.
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Step 6.2.1.1.5.1
Multiply by .
Step 6.2.1.1.5.2
Apply the distributive property.
Step 6.2.1.1.5.3
Multiply by .
Step 6.2.1.1.6
To write as a fraction with a common denominator, multiply by .
Step 6.2.1.1.7
Combine and .
Step 6.2.1.1.8
Combine the numerators over the common denominator.
Step 6.2.1.1.9
Rewrite using the commutative property of multiplication.
Step 6.2.1.1.10
Factor out of .
Step 6.2.1.1.11
Factor out of .
Step 6.2.1.1.12
Factor out of .
Step 6.2.1.1.13
Factor out of .
Step 6.2.1.1.14
Factor out of .
Step 6.2.1.1.15
Factor out of .
Step 6.2.1.1.16
Factor out of .
Step 6.2.1.1.17
Simplify the expression.
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Step 6.2.1.1.17.1
Rewrite as .
Step 6.2.1.1.17.2
Move the negative in front of the fraction.
Step 6.2.2
Simplify the right side.
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Step 6.2.2.1
Simplify .
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Step 6.2.2.1.1
Apply the distributive property.
Step 6.2.2.1.2
Multiply by by adding the exponents.
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Step 6.2.2.1.2.1
Move .
Step 6.2.2.1.2.2
Multiply by .
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Step 6.2.2.1.2.2.1
Raise to the power of .
Step 6.2.2.1.2.2.2
Use the power rule to combine exponents.
Step 6.2.2.1.2.3
Add and .
Step 6.2.2.1.3
Reorder factors in .
Step 6.2.2.1.4
Reorder and .
Step 6.3
Solve for .
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Step 6.3.1
Multiply both sides by .
Step 6.3.2
Simplify.
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Step 6.3.2.1
Simplify the left side.
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Step 6.3.2.1.1
Simplify .
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Step 6.3.2.1.1.1
Cancel the common factor of .
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Step 6.3.2.1.1.1.1
Move the leading negative in into the numerator.
Step 6.3.2.1.1.1.2
Cancel the common factor.
Step 6.3.2.1.1.1.3
Rewrite the expression.
Step 6.3.2.1.1.2
Apply the distributive property.
Step 6.3.2.1.1.3
Simplify.
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Step 6.3.2.1.1.3.1
Multiply by .
Step 6.3.2.1.1.3.2
Multiply .
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Step 6.3.2.1.1.3.2.1
Multiply by .
Step 6.3.2.1.1.3.2.2
Multiply by .
Step 6.3.2.1.1.3.3
Multiply .
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Step 6.3.2.1.1.3.3.1
Multiply by .
Step 6.3.2.1.1.3.3.2
Multiply by .
Step 6.3.2.1.1.3.4
Multiply by .
Step 6.3.2.1.1.4
Remove parentheses.
Step 6.3.2.1.1.5
Simplify the expression.
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Step 6.3.2.1.1.5.1
Move .
Step 6.3.2.1.1.5.2
Move .
Step 6.3.2.1.1.5.3
Reorder and .
Step 6.3.2.2
Simplify the right side.
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Step 6.3.2.2.1
Simplify .
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Step 6.3.2.2.1.1
Apply the distributive property.
Step 6.3.2.2.1.2
Reorder.
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Step 6.3.2.2.1.2.1
Rewrite using the commutative property of multiplication.
Step 6.3.2.2.1.2.2
Rewrite using the commutative property of multiplication.
Step 6.3.3
Solve for .
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Step 6.3.3.1
Subtract from both sides of the equation.
Step 6.3.3.2
Move all terms not containing to the right side of the equation.
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Step 6.3.3.2.1
Subtract from both sides of the equation.
Step 6.3.3.2.2
Subtract from both sides of the equation.
Step 6.3.3.3
Factor out of .
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Step 6.3.3.3.1
Factor out of .
Step 6.3.3.3.2
Factor out of .
Step 6.3.3.3.3
Factor out of .
Step 6.3.3.3.4
Factor out of .
Step 6.3.3.3.5
Factor out of .
Step 6.3.3.4
Divide each term in by and simplify.
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Step 6.3.3.4.1
Divide each term in by .
Step 6.3.3.4.2
Simplify the left side.
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Step 6.3.3.4.2.1
Cancel the common factor.
Step 6.3.3.4.2.2
Divide by .
Step 6.3.3.4.3
Simplify the right side.
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Step 6.3.3.4.3.1
Simplify each term.
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Step 6.3.3.4.3.1.1
Move the negative in front of the fraction.
Step 6.3.3.4.3.1.2
Move the negative in front of the fraction.
Step 6.3.3.4.3.2
Simplify terms.
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Step 6.3.3.4.3.2.1
Combine the numerators over the common denominator.
Step 6.3.3.4.3.2.2
Combine the numerators over the common denominator.
Step 6.3.3.4.3.2.3
Factor out of .
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Step 6.3.3.4.3.2.3.1
Factor out of .
Step 6.3.3.4.3.2.3.2
Factor out of .
Step 6.3.3.4.3.2.3.3
Factor out of .
Step 6.3.3.4.3.2.3.4
Factor out of .
Step 6.3.3.4.3.2.3.5
Factor out of .
Step 7
Replace with .