Calculus Examples

Find dy/dx x^2=(4x^2y^3+1)^2
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate using the Power Rule which states that is where .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the chain rule, which states that is where and .
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Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate.
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Step 3.2.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Differentiate using the Product Rule which states that is where and .
Step 3.4
Differentiate using the chain rule, which states that is where and .
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Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Move to the left of .
Step 3.6
Rewrite as .
Step 3.7
Differentiate using the Power Rule which states that is where .
Step 3.8
Move to the left of .
Step 3.9
Since is constant with respect to , the derivative of with respect to is .
Step 3.10
Simplify the expression.
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Step 3.10.1
Add and .
Step 3.10.2
Multiply by .
Step 3.11
Simplify.
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Step 3.11.1
Apply the distributive property.
Step 3.11.2
Combine terms.
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Step 3.11.2.1
Multiply by .
Step 3.11.2.2
Multiply by .
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
Rewrite the equation as .
Step 5.2
Divide each term in by and simplify.
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Step 5.2.1
Divide each term in by .
Step 5.2.2
Simplify the left side.
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Step 5.2.2.1
Cancel the common factor of .
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Step 5.2.2.1.1
Cancel the common factor.
Step 5.2.2.1.2
Divide by .
Step 5.2.3
Simplify the right side.
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Step 5.2.3.1
Cancel the common factor of and .
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Step 5.2.3.1.1
Factor out of .
Step 5.2.3.1.2
Cancel the common factors.
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Step 5.2.3.1.2.1
Factor out of .
Step 5.2.3.1.2.2
Factor out of .
Step 5.2.3.1.2.3
Factor out of .
Step 5.2.3.1.2.4
Cancel the common factor.
Step 5.2.3.1.2.5
Rewrite the expression.
Step 5.2.3.2
Factor out of .
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Step 5.2.3.2.1
Factor out of .
Step 5.2.3.2.2
Factor out of .
Step 5.2.3.2.3
Factor out of .
Step 5.3
Subtract from both sides of the equation.
Step 5.4
Divide each term in by and simplify.
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Step 5.4.1
Divide each term in by .
Step 5.4.2
Simplify the left side.
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Step 5.4.2.1
Cancel the common factor of .
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Step 5.4.2.1.1
Cancel the common factor.
Step 5.4.2.1.2
Rewrite the expression.
Step 5.4.2.2
Cancel the common factor of .
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Step 5.4.2.2.1
Cancel the common factor.
Step 5.4.2.2.2
Rewrite the expression.
Step 5.4.2.3
Cancel the common factor of .
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Step 5.4.2.3.1
Cancel the common factor.
Step 5.4.2.3.2
Divide by .
Step 5.4.3
Simplify the right side.
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Step 5.4.3.1
Combine the numerators over the common denominator.
Step 5.4.3.2
Simplify the numerator.
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Step 5.4.3.2.1
Factor out of .
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Step 5.4.3.2.1.1
Factor out of .
Step 5.4.3.2.1.2
Factor out of .
Step 5.4.3.2.1.3
Factor out of .
Step 5.4.3.2.2
To write as a fraction with a common denominator, multiply by .
Step 5.4.3.2.3
Combine and .
Step 5.4.3.2.4
Combine the numerators over the common denominator.
Step 5.4.3.2.5
Simplify the numerator.
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Step 5.4.3.2.5.1
Apply the distributive property.
Step 5.4.3.2.5.2
Multiply by .
Step 5.4.3.2.5.3
Multiply by .
Step 5.4.3.2.5.4
Apply the distributive property.
Step 5.4.3.2.5.5
Multiply by by adding the exponents.
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Step 5.4.3.2.5.5.1
Move .
Step 5.4.3.2.5.5.2
Use the power rule to combine exponents.
Step 5.4.3.2.5.5.3
Add and .
Step 5.4.3.2.5.6
Multiply by .
Step 5.4.3.2.5.7
Simplify each term.
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Step 5.4.3.2.5.7.1
Rewrite using the commutative property of multiplication.
Step 5.4.3.2.5.7.2
Multiply by .
Step 5.4.3.3
Combine and .
Step 5.4.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 5.4.3.5
Simplify terms.
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Step 5.4.3.5.1
Combine.
Step 5.4.3.5.2
Cancel the common factor of and .
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Step 5.4.3.5.2.1
Factor out of .
Step 5.4.3.5.2.2
Cancel the common factors.
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Step 5.4.3.5.2.2.1
Factor out of .
Step 5.4.3.5.2.2.2
Cancel the common factor.
Step 5.4.3.5.2.2.3
Rewrite the expression.
Step 5.4.3.5.3
Multiply.
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Step 5.4.3.5.3.1
Multiply by .
Step 5.4.3.5.3.2
Multiply by .
Step 5.4.3.6
Simplify the denominator.
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Step 5.4.3.6.1
Rewrite.
Step 5.4.3.6.2
Factor out of .
Step 5.4.3.6.3
Rewrite.
Step 5.4.3.6.4
Simplify.
Step 5.4.3.6.5
Remove unnecessary parentheses.
Step 5.4.3.7
Reorder factors in .
Step 6
Replace with .