Calculus Examples

Find dy/dx arctan(2x^2y)=x+3xy^2
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
Differentiate using the chain rule, which states that is where and .
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Step 2.1.1
To apply the Chain Rule, set as .
Step 2.1.2
The derivative of with respect to is .
Step 2.1.3
Replace all occurrences of with .
Step 2.2
Differentiate using the Constant Multiple Rule.
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Step 2.2.1
Factor out of .
Step 2.2.2
Simplify the expression.
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Step 2.2.2.1
Apply the product rule to .
Step 2.2.2.2
Raise to the power of .
Step 2.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.4
Combine and .
Step 2.3
Differentiate using the Product Rule which states that is where and .
Step 2.4
Rewrite as .
Step 2.5
Differentiate using the Power Rule which states that is where .
Step 2.6
Move to the left of .
Step 2.7
Simplify.
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Step 2.7.1
Apply the product rule to .
Step 2.7.2
Multiply the exponents in .
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Step 2.7.2.1
Apply the power rule and multiply exponents, .
Step 2.7.2.2
Multiply by .
Step 2.7.3
Reorder the factors of .
Step 2.7.4
Multiply by .
Step 2.7.5
Factor out of .
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Step 2.7.5.1
Factor out of .
Step 2.7.5.2
Factor out of .
Step 2.7.5.3
Factor out of .
Step 2.7.6
Move to the left of .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate.
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Step 3.1.1
By the Sum Rule, the derivative of with respect to is .
Step 3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.2
Evaluate .
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Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Differentiate using the Product Rule which states that is where and .
Step 3.2.3
Differentiate using the chain rule, which states that is where and .
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Step 3.2.3.1
To apply the Chain Rule, set as .
Step 3.2.3.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3.3
Replace all occurrences of with .
Step 3.2.4
Rewrite as .
Step 3.2.5
Differentiate using the Power Rule which states that is where .
Step 3.2.6
Move to the left of .
Step 3.2.7
Multiply by .
Step 3.3
Simplify.
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Step 3.3.1
Apply the distributive property.
Step 3.3.2
Multiply by .
Step 3.3.3
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Solve for .
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Step 5.1
Multiply both sides by .
Step 5.2
Simplify.
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Step 5.2.1
Simplify the left side.
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Step 5.2.1.1
Simplify .
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Step 5.2.1.1.1
Cancel the common factor of .
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Step 5.2.1.1.1.1
Cancel the common factor.
Step 5.2.1.1.1.2
Rewrite the expression.
Step 5.2.1.1.2
Apply the distributive property.
Step 5.2.1.1.3
Multiply by by adding the exponents.
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Step 5.2.1.1.3.1
Move .
Step 5.2.1.1.3.2
Multiply by .
Step 5.2.1.1.4
Rewrite using the commutative property of multiplication.
Step 5.2.1.1.5
Multiply by .
Step 5.2.1.1.6
Move .
Step 5.2.2
Simplify the right side.
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Step 5.2.2.1
Simplify .
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Step 5.2.2.1.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.2.2.1.2
Simplify terms.
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Step 5.2.2.1.2.1
Simplify each term.
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Step 5.2.2.1.2.1.1
Multiply by .
Step 5.2.2.1.2.1.2
Multiply by by adding the exponents.
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Step 5.2.2.1.2.1.2.1
Move .
Step 5.2.2.1.2.1.2.2
Use the power rule to combine exponents.
Step 5.2.2.1.2.1.2.3
Add and .
Step 5.2.2.1.2.1.3
Rewrite using the commutative property of multiplication.
Step 5.2.2.1.2.1.4
Multiply by .
Step 5.2.2.1.2.1.5
Multiply by .
Step 5.2.2.1.2.1.6
Multiply by by adding the exponents.
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Step 5.2.2.1.2.1.6.1
Move .
Step 5.2.2.1.2.1.6.2
Multiply by .
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Step 5.2.2.1.2.1.6.2.1
Raise to the power of .
Step 5.2.2.1.2.1.6.2.2
Use the power rule to combine exponents.
Step 5.2.2.1.2.1.6.3
Add and .
Step 5.2.2.1.2.1.7
Multiply by by adding the exponents.
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Step 5.2.2.1.2.1.7.1
Move .
Step 5.2.2.1.2.1.7.2
Multiply by .
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Step 5.2.2.1.2.1.7.2.1
Raise to the power of .
Step 5.2.2.1.2.1.7.2.2
Use the power rule to combine exponents.
Step 5.2.2.1.2.1.7.3
Add and .
Step 5.2.2.1.2.1.8
Multiply by .
Step 5.2.2.1.2.1.9
Multiply by .
Step 5.2.2.1.2.1.10
Multiply by .
Step 5.2.2.1.2.2
Simplify the expression.
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Step 5.2.2.1.2.2.1
Move .
Step 5.2.2.1.2.2.2
Move .
Step 5.2.2.1.2.2.3
Move .
Step 5.2.2.1.2.2.4
Move .
Step 5.2.2.1.2.2.5
Move .
Step 5.2.2.1.2.2.6
Move .
Step 5.2.2.1.2.2.7
Move .
Step 5.2.2.1.2.2.8
Move .
Step 5.2.2.1.2.2.9
Reorder and .
Step 5.3
Solve for .
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Step 5.3.1
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 5.3.2
Subtract from both sides of the equation.
Step 5.3.3
Move all terms not containing to the right side of the equation.
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Step 5.3.3.1
Subtract from both sides of the equation.
Step 5.3.3.2
Subtract from both sides of the equation.
Step 5.3.3.3
Subtract from both sides of the equation.
Step 5.3.3.4
Subtract from both sides of the equation.
Step 5.3.4
Factor out of .
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Step 5.3.4.1
Factor out of .
Step 5.3.4.2
Factor out of .
Step 5.3.4.3
Factor out of .
Step 5.3.4.4
Factor out of .
Step 5.3.4.5
Factor out of .
Step 5.3.5
Rewrite as .
Step 5.3.6
Divide each term in by and simplify.
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Step 5.3.6.1
Divide each term in by .
Step 5.3.6.2
Simplify the left side.
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Step 5.3.6.2.1
Cancel the common factor of .
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Step 5.3.6.2.1.1
Cancel the common factor.
Step 5.3.6.2.1.2
Rewrite the expression.
Step 5.3.6.2.2
Cancel the common factor of .
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Step 5.3.6.2.2.1
Cancel the common factor.
Step 5.3.6.2.2.2
Rewrite the expression.
Step 5.3.6.2.3
Cancel the common factor of .
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Step 5.3.6.2.3.1
Cancel the common factor.
Step 5.3.6.2.3.2
Divide by .
Step 5.3.6.3
Simplify the right side.
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Step 5.3.6.3.1
Simplify each term.
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Step 5.3.6.3.1.1
Cancel the common factor of and .
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Step 5.3.6.3.1.1.1
Factor out of .
Step 5.3.6.3.1.1.2
Cancel the common factors.
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Step 5.3.6.3.1.1.2.1
Factor out of .
Step 5.3.6.3.1.1.2.2
Cancel the common factor.
Step 5.3.6.3.1.1.2.3
Rewrite the expression.
Step 5.3.6.3.1.2
Cancel the common factor of .
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Step 5.3.6.3.1.2.1
Cancel the common factor.
Step 5.3.6.3.1.2.2
Rewrite the expression.
Step 5.3.6.3.1.3
Cancel the common factor of and .
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Step 5.3.6.3.1.3.1
Factor out of .
Step 5.3.6.3.1.3.2
Cancel the common factors.
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Step 5.3.6.3.1.3.2.1
Factor out of .
Step 5.3.6.3.1.3.2.2
Cancel the common factor.
Step 5.3.6.3.1.3.2.3
Rewrite the expression.
Step 5.3.6.3.1.4
Cancel the common factor of and .
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Step 5.3.6.3.1.4.1
Factor out of .
Step 5.3.6.3.1.4.2
Cancel the common factors.
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Step 5.3.6.3.1.4.2.1
Cancel the common factor.
Step 5.3.6.3.1.4.2.2
Rewrite the expression.
Step 5.3.6.3.1.5
Move the negative in front of the fraction.
Step 5.3.6.3.1.6
Cancel the common factor of and .
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Step 5.3.6.3.1.6.1
Factor out of .
Step 5.3.6.3.1.6.2
Cancel the common factors.
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Step 5.3.6.3.1.6.2.1
Factor out of .
Step 5.3.6.3.1.6.2.2
Cancel the common factor.
Step 5.3.6.3.1.6.2.3
Rewrite the expression.
Step 5.3.6.3.1.7
Cancel the common factor of and .
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Step 5.3.6.3.1.7.1
Factor out of .
Step 5.3.6.3.1.7.2
Cancel the common factors.
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Step 5.3.6.3.1.7.2.1
Cancel the common factor.
Step 5.3.6.3.1.7.2.2
Rewrite the expression.
Step 5.3.6.3.1.8
Move the negative in front of the fraction.
Step 5.3.6.3.1.9
Move the negative in front of the fraction.
Step 5.3.6.3.1.10
Move the negative in front of the fraction.
Step 5.3.6.3.2
Simplify terms.
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Step 5.3.6.3.2.1
Combine the numerators over the common denominator.
Step 5.3.6.3.2.2
Factor out of .
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Step 5.3.6.3.2.2.1
Factor out of .
Step 5.3.6.3.2.2.2
Factor out of .
Step 5.3.6.3.2.2.3
Factor out of .
Step 5.3.6.3.2.3
Combine the numerators over the common denominator.
Step 5.3.6.3.2.4
Factor out of .
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Step 5.3.6.3.2.4.1
Factor out of .
Step 5.3.6.3.2.4.2
Factor out of .
Step 5.3.6.3.3
To write as a fraction with a common denominator, multiply by .
Step 5.3.6.3.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 5.3.6.3.4.1
Multiply by .
Step 5.3.6.3.4.2
Reorder the factors of .
Step 5.3.6.3.5
Combine the numerators over the common denominator.
Step 5.3.6.3.6
Simplify the numerator.
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Step 5.3.6.3.6.1
Factor out of .
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Step 5.3.6.3.6.1.1
Factor out of .
Step 5.3.6.3.6.1.2
Factor out of .
Step 5.3.6.3.6.1.3
Factor out of .
Step 5.3.6.3.6.2
Apply the distributive property.
Step 5.3.6.3.6.3
Simplify.
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Step 5.3.6.3.6.3.1
Multiply by .
Step 5.3.6.3.6.3.2
Multiply by .
Step 5.3.6.3.6.3.3
Multiply by .
Step 5.3.6.3.6.4
Remove parentheses.
Step 5.3.6.3.6.5
Apply the distributive property.
Step 5.3.6.3.6.6
Simplify.
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Step 5.3.6.3.6.6.1
Multiply by .
Step 5.3.6.3.6.6.2
Multiply by by adding the exponents.
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Step 5.3.6.3.6.6.2.1
Move .
Step 5.3.6.3.6.6.2.2
Multiply by .
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Step 5.3.6.3.6.6.2.2.1
Raise to the power of .
Step 5.3.6.3.6.6.2.2.2
Use the power rule to combine exponents.
Step 5.3.6.3.6.6.2.3
Add and .
Step 5.3.6.3.6.6.3
Multiply by by adding the exponents.
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Step 5.3.6.3.6.6.3.1
Move .
Step 5.3.6.3.6.6.3.2
Multiply by .
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Step 5.3.6.3.6.6.3.2.1
Raise to the power of .
Step 5.3.6.3.6.6.3.2.2
Use the power rule to combine exponents.
Step 5.3.6.3.6.6.3.3
Add and .
Step 5.3.6.3.6.7
Simplify each term.
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Step 5.3.6.3.6.7.1
Multiply by .
Step 5.3.6.3.6.7.2
Multiply by .
Step 5.3.6.3.7
Combine the numerators over the common denominator.
Step 5.3.6.3.8
Simplify the numerator.
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Step 5.3.6.3.8.1
Apply the distributive property.
Step 5.3.6.3.8.2
Simplify.
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Step 5.3.6.3.8.2.1
Rewrite using the commutative property of multiplication.
Step 5.3.6.3.8.2.2
Rewrite using the commutative property of multiplication.
Step 5.3.6.3.8.2.3
Rewrite using the commutative property of multiplication.
Step 5.3.6.3.8.2.4
Rewrite using the commutative property of multiplication.
Step 5.3.6.3.8.3
Simplify each term.
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Step 5.3.6.3.8.3.1
Multiply by by adding the exponents.
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Step 5.3.6.3.8.3.1.1
Move .
Step 5.3.6.3.8.3.1.2
Multiply by .
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Step 5.3.6.3.8.3.1.2.1
Raise to the power of .
Step 5.3.6.3.8.3.1.2.2
Use the power rule to combine exponents.
Step 5.3.6.3.8.3.1.3
Add and .
Step 5.3.6.3.8.3.2
Multiply by by adding the exponents.
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Step 5.3.6.3.8.3.2.1
Move .
Step 5.3.6.3.8.3.2.2
Multiply by .
Step 5.3.6.3.8.3.3
Multiply by by adding the exponents.
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Step 5.3.6.3.8.3.3.1
Move .
Step 5.3.6.3.8.3.3.2
Multiply by .
Step 6
Replace with .