Calculus Examples

Find dy/dx natural log of 9y=e^ysin(5x)
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
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Simplify terms.
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Step 2.2.2.1
Combine and .
Step 2.2.2.2
Cancel the common factor of .
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Step 2.2.2.2.1
Cancel the common factor.
Step 2.2.2.2.2
Rewrite the expression.
Step 2.3
Rewrite as .
Step 2.4
Combine and .
Step 3
Differentiate the right 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
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate.
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Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Simplify the expression.
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Step 3.3.3.1
Multiply by .
Step 3.3.3.2
Move to the left of .
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 Exponential Rule which states that is where =.
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Rewrite as .
Step 3.6
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
Cancel the common factor of .
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Step 5.2.1.1.1
Cancel the common factor.
Step 5.2.1.1.2
Rewrite the expression.
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
Apply the distributive property.
Step 5.2.2.1.2
Simplify the expression.
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Step 5.2.2.1.2.1
Reorder factors in .
Step 5.2.2.1.2.2
Reorder and .
Step 5.3
Solve for .
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Step 5.3.1
Subtract from both sides of the equation.
Step 5.3.2
Factor out of .
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Step 5.3.2.1
Factor out of .
Step 5.3.2.2
Factor out of .
Step 5.3.2.3
Factor out of .
Step 5.3.3
Rewrite as .
Step 5.3.4
Divide each term in by and simplify.
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Step 5.3.4.1
Divide each term in by .
Step 5.3.4.2
Simplify the left side.
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Step 5.3.4.2.1
Cancel the common factor of .
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Step 5.3.4.2.1.1
Cancel the common factor.
Step 5.3.4.2.1.2
Divide by .
Step 6
Replace with .