Calculus Examples

Find dy/dx x^3 = natural log of xy
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
The derivative of with respect to is .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate using the Product Rule which states that is where and .
Step 3.3
Rewrite as .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Multiply by .
Step 3.6
Simplify.
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Step 3.6.1
Apply the distributive property.
Step 3.6.2
Combine terms.
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Step 3.6.2.1
Combine and .
Step 3.6.2.2
Combine and .
Step 3.6.2.3
Cancel the common factor of .
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Step 3.6.2.3.1
Cancel the common factor.
Step 3.6.2.3.2
Rewrite the expression.
Step 3.6.2.4
Combine and .
Step 3.6.2.5
Cancel the common factor of .
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Step 3.6.2.5.1
Cancel the common factor.
Step 3.6.2.5.2
Rewrite the expression.
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
Subtract from both sides of the equation.
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
Cancel the common factor of .
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Step 5.4.1.1.1
Cancel the common factor.
Step 5.4.1.1.2
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
Apply the distributive property.
Step 5.4.2.1.2
Combine and .
Step 6
Replace with .