Calculus Examples

Find the Derivative - d/dx natural log of csc(x)
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
Rewrite in terms of sines and cosines.
Step 3
Multiply by the reciprocal of the fraction to divide by .
Step 4
Multiply by .
Step 5
The derivative of with respect to is .
Step 6
Simplify.
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Step 6.1
Reorder the factors of .
Step 6.2
Rewrite in terms of sines and cosines.
Step 6.3
Rewrite in terms of sines and cosines.
Step 6.4
Multiply .
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Step 6.4.1
Multiply by .
Step 6.4.2
Raise to the power of .
Step 6.4.3
Raise to the power of .
Step 6.4.4
Use the power rule to combine exponents.
Step 6.4.5
Add and .
Step 6.5
Cancel the common factor of .
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Step 6.5.1
Move the leading negative in into the numerator.
Step 6.5.2
Factor out of .
Step 6.5.3
Cancel the common factor.
Step 6.5.4
Rewrite the expression.
Step 6.6
Move the negative in front of the fraction.
Step 6.7
Convert from to .