Calculus Examples

Find the Derivative - d/dx arccsc(x^2+1)
Step 1
Differentiate using the chain rule, which states that is where and .
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Step 1.1
To apply the Chain Rule, set as .
Step 1.2
The derivative of with respect to is .
Step 1.3
Replace all occurrences of with .
Step 2
Differentiate.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Combine fractions.
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Step 2.4.1
Add and .
Step 2.4.2
Multiply by .
Step 2.4.3
Combine and .
Step 2.4.4
Combine and .
Step 2.4.5
Move the negative in front of the fraction.
Step 3
Simplify.
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Step 3.1
Simplify the denominator.
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Step 3.1.1
Rewrite as .
Step 3.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.1.3
Simplify.
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Step 3.1.3.1
Add and .
Step 3.1.3.2
Subtract from .
Step 3.1.3.3
Add and .
Step 3.1.4
Reorder and .
Step 3.1.5
Pull terms out from under the radical.
Step 3.2
Cancel the common factor of .
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Step 3.2.1
Cancel the common factor.
Step 3.2.2
Rewrite the expression.
Step 3.3
Multiply by .
Step 3.4
Combine and simplify the denominator.
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Step 3.4.1
Multiply by .
Step 3.4.2
Move .
Step 3.4.3
Raise to the power of .
Step 3.4.4
Raise to the power of .
Step 3.4.5
Use the power rule to combine exponents.
Step 3.4.6
Add and .
Step 3.4.7
Rewrite as .
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Step 3.4.7.1
Use to rewrite as .
Step 3.4.7.2
Apply the power rule and multiply exponents, .
Step 3.4.7.3
Combine and .
Step 3.4.7.4
Cancel the common factor of .
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Step 3.4.7.4.1
Cancel the common factor.
Step 3.4.7.4.2
Rewrite the expression.
Step 3.4.7.5
Simplify.