Calculus Examples

Find dy/dx y=(sec(2x))/(1+tan(2x))
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the Quotient 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 3.3.4
By the Sum Rule, the derivative of with respect to is .
Step 3.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.6
Add 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
The derivative of with respect to is .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Raise to the power of .
Step 3.6
Use the power rule to combine exponents.
Step 3.7
Add and .
Step 3.8
Since is constant with respect to , the derivative of with respect to is .
Step 3.9
Multiply by .
Step 3.10
Differentiate using the Power Rule which states that is where .
Step 3.11
Multiply by .
Step 3.12
Simplify.
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Step 3.12.1
Apply the distributive property.
Step 3.12.2
Apply the distributive property.
Step 3.12.3
Apply the distributive property.
Step 3.12.4
Simplify the numerator.
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Step 3.12.4.1
Simplify each term.
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Step 3.12.4.1.1
Multiply by .
Step 3.12.4.1.2
Multiply .
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Step 3.12.4.1.2.1
Raise to the power of .
Step 3.12.4.1.2.2
Raise to the power of .
Step 3.12.4.1.2.3
Use the power rule to combine exponents.
Step 3.12.4.1.2.4
Add and .
Step 3.12.4.2
Factor out of .
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Step 3.12.4.2.1
Factor out of .
Step 3.12.4.2.2
Factor out of .
Step 3.12.4.2.3
Factor out of .
Step 3.12.4.2.4
Factor out of .
Step 3.12.4.2.5
Factor out of .
Step 3.12.4.3
Reorder and .
Step 3.12.4.4
Factor out of .
Step 3.12.4.5
Factor out of .
Step 3.12.4.6
Factor out of .
Step 3.12.4.7
Apply pythagorean identity.
Step 3.12.4.8
Multiply by .
Step 3.12.4.9
Apply the distributive property.
Step 3.12.4.10
Multiply by .
Step 3.12.5
Factor out of .
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Step 3.12.5.1
Factor out of .
Step 3.12.5.2
Factor out of .
Step 3.12.5.3
Factor out of .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .