Calculus Examples

Find the Derivative - d/dx y=(csc(x)+cot(x))(csc(x)-cot(x))
Step 1
Differentiate using the Product Rule which states that is where and .
Step 2
By the Sum Rule, the derivative of with respect to is .
Step 3
The derivative of with respect to is .
Step 4
Since is constant with respect to , the derivative of with respect to is .
Step 5
The derivative of with respect to is .
Step 6
Differentiate using the Sum Rule.
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Step 6.1
Multiply by .
Step 6.2
Multiply by .
Step 6.3
By the Sum Rule, the derivative of with respect to is .
Step 7
The derivative of with respect to is .
Step 8
The derivative of with respect to is .
Step 9
Simplify.
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Step 9.1
Reorder terms.
Step 9.2
Simplify each term.
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Step 9.2.1
Expand using the FOIL Method.
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Step 9.2.1.1
Apply the distributive property.
Step 9.2.1.2
Apply the distributive property.
Step 9.2.1.3
Apply the distributive property.
Step 9.2.2
Simplify and combine like terms.
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Step 9.2.2.1
Simplify each term.
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Step 9.2.2.1.1
Multiply .
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Step 9.2.2.1.1.1
Raise to the power of .
Step 9.2.2.1.1.2
Raise to the power of .
Step 9.2.2.1.1.3
Use the power rule to combine exponents.
Step 9.2.2.1.1.4
Add and .
Step 9.2.2.1.2
Multiply .
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Step 9.2.2.1.2.1
Raise to the power of .
Step 9.2.2.1.2.2
Raise to the power of .
Step 9.2.2.1.2.3
Use the power rule to combine exponents.
Step 9.2.2.1.2.4
Add and .
Step 9.2.2.1.3
Multiply by by adding the exponents.
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Step 9.2.2.1.3.1
Multiply by .
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Step 9.2.2.1.3.1.1
Raise to the power of .
Step 9.2.2.1.3.1.2
Use the power rule to combine exponents.
Step 9.2.2.1.3.2
Add and .
Step 9.2.2.2
Add and .
Step 9.2.2.3
Subtract from .
Step 9.2.3
Factor out of .
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Step 9.2.3.1
Factor out of .
Step 9.2.3.2
Factor out of .
Step 9.2.3.3
Factor out of .
Step 9.2.4
Reorder and .
Step 9.2.5
Apply pythagorean identity.
Step 9.2.6
Multiply by .
Step 9.2.7
Expand using the FOIL Method.
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Step 9.2.7.1
Apply the distributive property.
Step 9.2.7.2
Apply the distributive property.
Step 9.2.7.3
Apply the distributive property.
Step 9.2.8
Simplify and combine like terms.
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Step 9.2.8.1
Simplify each term.
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Step 9.2.8.1.1
Multiply .
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Step 9.2.8.1.1.1
Raise to the power of .
Step 9.2.8.1.1.2
Raise to the power of .
Step 9.2.8.1.1.3
Use the power rule to combine exponents.
Step 9.2.8.1.1.4
Add and .
Step 9.2.8.1.2
Multiply .
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Step 9.2.8.1.2.1
Multiply by .
Step 9.2.8.1.2.2
Multiply by .
Step 9.2.8.1.2.3
Raise to the power of .
Step 9.2.8.1.2.4
Raise to the power of .
Step 9.2.8.1.2.5
Use the power rule to combine exponents.
Step 9.2.8.1.2.6
Add and .
Step 9.2.8.1.3
Multiply by by adding the exponents.
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Step 9.2.8.1.3.1
Move .
Step 9.2.8.1.3.2
Multiply by .
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Step 9.2.8.1.3.2.1
Raise to the power of .
Step 9.2.8.1.3.2.2
Use the power rule to combine exponents.
Step 9.2.8.1.3.3
Add and .
Step 9.2.8.1.4
Multiply .
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Step 9.2.8.1.4.1
Multiply by .
Step 9.2.8.1.4.2
Multiply by .
Step 9.2.8.2
Add and .
Step 9.2.8.3
Add and .
Step 9.2.9
Factor out of .
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Step 9.2.9.1
Factor out of .
Step 9.2.9.2
Factor out of .
Step 9.2.9.3
Factor out of .
Step 9.2.10
Reorder and .
Step 9.2.11
Factor out of .
Step 9.2.12
Factor out of .
Step 9.2.13
Factor out of .
Step 9.2.14
Apply pythagorean identity.
Step 9.2.15
Multiply by .
Step 9.2.16
Move to the left of .
Step 9.2.17
Rewrite as .
Step 9.3
Subtract from .