Calculus Examples

Find dy/dx x^(1/3)-y^(1/3)=1
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Differentiate using the Power Rule which states that is where .
Step 2.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.2.3
Combine and .
Step 2.2.4
Combine the numerators over the common denominator.
Step 2.2.5
Simplify the numerator.
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Step 2.2.5.1
Multiply by .
Step 2.2.5.2
Subtract from .
Step 2.2.6
Move the negative in front of the fraction.
Step 2.3
Evaluate .
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Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the chain rule, which states that is where and .
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Step 2.3.2.1
To apply the Chain Rule, set as .
Step 2.3.2.2
Differentiate using the Power Rule which states that is where .
Step 2.3.2.3
Replace all occurrences of with .
Step 2.3.3
Rewrite as .
Step 2.3.4
To write as a fraction with a common denominator, multiply by .
Step 2.3.5
Combine and .
Step 2.3.6
Combine the numerators over the common denominator.
Step 2.3.7
Simplify the numerator.
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Step 2.3.7.1
Multiply by .
Step 2.3.7.2
Subtract from .
Step 2.3.8
Move the negative in front of the fraction.
Step 2.3.9
Combine and .
Step 2.3.10
Combine and .
Step 2.3.11
Move to the denominator using the negative exponent rule .
Step 2.4
Simplify.
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Step 2.4.1
Rewrite the expression using the negative exponent rule .
Step 2.4.2
Multiply by .
Step 3
Since is constant with respect to , the derivative of with respect to is .
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
Subtract from both sides of the equation.
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
Dividing two negative values results in a positive value.
Step 5.2.2.2
Divide by .
Step 5.2.3
Simplify the right side.
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Step 5.2.3.1
Dividing two negative values results in a positive value.
Step 5.2.3.2
Divide by .
Step 5.3
Multiply both sides by .
Step 5.4
Simplify.
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Step 5.4.1
Simplify the left side.
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Step 5.4.1.1
Simplify .
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Step 5.4.1.1.1
Rewrite using the commutative property of multiplication.
Step 5.4.1.1.2
Cancel the common factor of .
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Step 5.4.1.1.2.1
Cancel the common factor.
Step 5.4.1.1.2.2
Rewrite the expression.
Step 5.4.1.1.3
Cancel the common factor of .
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Step 5.4.1.1.3.1
Cancel the common factor.
Step 5.4.1.1.3.2
Rewrite the expression.
Step 5.4.2
Simplify the right side.
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Step 5.4.2.1
Simplify .
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Step 5.4.2.1.1
Rewrite using the commutative property of multiplication.
Step 5.4.2.1.2
Cancel the common factor of .
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Step 5.4.2.1.2.1
Cancel the common factor.
Step 5.4.2.1.2.2
Rewrite the expression.
Step 5.4.2.1.3
Combine and .
Step 6
Replace with .