Calculus Examples

Find dy/dx x square root of y+1=xy+1
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
Differentiate using the Product Rule which states that is where and .
Step 3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
To write as a fraction with a common denominator, multiply by .
Step 3.4
Combine and .
Step 3.5
Combine the numerators over the common denominator.
Step 3.6
Simplify the numerator.
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Step 3.6.1
Multiply by .
Step 3.6.2
Subtract from .
Step 3.7
Combine fractions.
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Step 3.7.1
Move the negative in front of the fraction.
Step 3.7.2
Combine and .
Step 3.7.3
Move to the denominator using the negative exponent rule .
Step 3.7.4
Combine and .
Step 3.8
By the Sum Rule, the derivative of with respect to is .
Step 3.9
Rewrite as .
Step 3.10
Since is constant with respect to , the derivative of with respect to is .
Step 3.11
Combine fractions.
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Step 3.11.1
Add and .
Step 3.11.2
Combine and .
Step 3.12
Differentiate using the Power Rule which states that is where .
Step 3.13
Multiply by .
Step 3.14
To write as a fraction with a common denominator, multiply by .
Step 3.15
Combine and .
Step 3.16
Combine the numerators over the common denominator.
Step 3.17
Multiply by by adding the exponents.
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Step 3.17.1
Move .
Step 3.17.2
Use the power rule to combine exponents.
Step 3.17.3
Combine the numerators over the common denominator.
Step 3.17.4
Add and .
Step 3.17.5
Divide by .
Step 3.18
Simplify .
Step 3.19
Move to the left of .
Step 3.20
Simplify.
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Step 3.20.1
Apply the distributive property.
Step 3.20.2
Multiply by .
Step 4
Differentiate the right side of the equation.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Evaluate .
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Step 4.2.1
Differentiate using the Product Rule which states that is where and .
Step 4.2.2
Rewrite as .
Step 4.2.3
Differentiate using the Power Rule which states that is where .
Step 4.2.4
Multiply by .
Step 4.3
Differentiate using the Constant Rule.
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Step 4.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.2
Add and .
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
Rewrite using the commutative property of multiplication.
Step 6.2.1.1.2
Cancel the common factor of .
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Step 6.2.1.1.2.1
Cancel the common factor.
Step 6.2.1.1.2.2
Rewrite the expression.
Step 6.2.1.1.3
Cancel the common factor of .
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Step 6.2.1.1.3.1
Cancel the common factor.
Step 6.2.1.1.3.2
Rewrite the expression.
Step 6.2.1.1.4
Reorder and .
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
Reorder.
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Step 6.2.2.1.2.1
Rewrite using the commutative property of multiplication.
Step 6.2.2.1.2.2
Rewrite using the commutative property of multiplication.
Step 6.2.2.1.2.3
Move .
Step 6.3
Solve for .
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Step 6.3.1
Subtract from both sides of the equation.
Step 6.3.2
Move all terms not containing to the right side of the equation.
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Step 6.3.2.1
Subtract from both sides of the equation.
Step 6.3.2.2
Subtract from both sides of the equation.
Step 6.3.3
Factor out of .
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Step 6.3.3.1
Factor out of .
Step 6.3.3.2
Factor out of .
Step 6.3.3.3
Factor out of .
Step 6.3.4
Divide each term in by and simplify.
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Step 6.3.4.1
Divide each term in by .
Step 6.3.4.2
Simplify the left side.
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Step 6.3.4.2.1
Cancel the common factor.
Step 6.3.4.2.2
Rewrite the expression.
Step 6.3.4.2.3
Cancel the common factor.
Step 6.3.4.2.4
Divide by .
Step 6.3.4.3
Simplify the right side.
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Step 6.3.4.3.1
Simplify terms.
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Step 6.3.4.3.1.1
Simplify each term.
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Step 6.3.4.3.1.1.1
Move the negative in front of the fraction.
Step 6.3.4.3.1.1.2
Move the negative in front of the fraction.
Step 6.3.4.3.1.2
Simplify terms.
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Step 6.3.4.3.1.2.1
Combine the numerators over the common denominator.
Step 6.3.4.3.1.2.2
Factor out of .
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Step 6.3.4.3.1.2.2.1
Factor out of .
Step 6.3.4.3.1.2.2.2
Factor out of .
Step 6.3.4.3.1.2.2.3
Factor out of .
Step 6.3.4.3.1.2.3
Combine the numerators over the common denominator.
Step 6.3.4.3.2
Simplify the numerator.
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Step 6.3.4.3.2.1
Factor out of .
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Step 6.3.4.3.2.1.1
Factor out of .
Step 6.3.4.3.2.1.2
Factor out of .
Step 6.3.4.3.2.1.3
Factor out of .
Step 6.3.4.3.2.2
Apply the distributive property.
Step 6.3.4.3.2.3
Move to the left of .
Step 6.3.4.3.2.4
Rewrite as .
Step 7
Replace with .