Calculus Examples

Find dy/dx 4cos(x)sin(y)=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
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Product Rule which states that is where and .
Step 2.3
Differentiate using the chain rule, which states that is where and .
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Step 2.3.1
To apply the Chain Rule, set as .
Step 2.3.2
The derivative of with respect to is .
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Rewrite as .
Step 2.5
The derivative of with respect to is .
Step 2.6
Simplify.
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Step 2.6.1
Apply the distributive property.
Step 2.6.2
Multiply by .
Step 2.6.3
Reorder terms.
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
Simplify the left side.
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Step 5.1.1
Reorder factors in .
Step 5.2
Add to both sides of the equation.
Step 5.3
Divide each term in by and simplify.
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Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
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Step 5.3.2.1
Cancel the common factor of .
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Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Rewrite the expression.
Step 5.3.2.2
Cancel the common factor of .
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Step 5.3.2.2.1
Cancel the common factor.
Step 5.3.2.2.2
Rewrite the expression.
Step 5.3.2.3
Cancel the common factor of .
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Step 5.3.2.3.1
Cancel the common factor.
Step 5.3.2.3.2
Divide by .
Step 5.3.3
Simplify the right side.
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Step 5.3.3.1
Cancel the common factor of .
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Step 5.3.3.1.1
Cancel the common factor.
Step 5.3.3.1.2
Rewrite the expression.
Step 5.3.3.2
Separate fractions.
Step 5.3.3.3
Convert from to .
Step 5.3.3.4
Convert from to .
Step 6
Replace with .