Calculus Examples

Find the Derivative - d/dx f(x) = natural log of tan(x/2+pi/4)
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
Convert from to .
Step 5
Differentiate using the chain rule, which states that is where and .
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Step 5.1
To apply the Chain Rule, set as .
Step 5.2
The derivative of with respect to is .
Step 5.3
Replace all occurrences of with .
Step 6
Differentiate.
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Step 6.1
By the Sum Rule, the derivative of with respect to is .
Step 6.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.3
Differentiate using the Power Rule which states that is where .
Step 6.4
Multiply by .
Step 6.5
Since is constant with respect to , the derivative of with respect to is .
Step 6.6
Combine fractions.
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Step 6.6.1
Add and .
Step 6.6.2
Combine and .
Step 6.6.3
Combine and .
Step 7
Simplify.
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Step 7.1
Simplify the numerator.
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Step 7.1.1
Rewrite in terms of sines and cosines.
Step 7.1.2
Rewrite in terms of sines and cosines.
Step 7.1.3
Apply the product rule to .
Step 7.1.4
Cancel the common factor of .
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Step 7.1.4.1
Factor out of .
Step 7.1.4.2
Cancel the common factor.
Step 7.1.4.3
Rewrite the expression.
Step 7.1.5
Multiply by .
Step 7.1.6
Separate fractions.
Step 7.1.7
Convert from to .
Step 7.1.8
Separate fractions.
Step 7.1.9
Convert from to .
Step 7.1.10
Divide by .
Step 7.1.11
One to any power is one.
Step 7.1.12
Multiply by .
Step 7.2
Reorder terms.