Calculus Examples

Find the Derivative - d/dx y=sin(x)^(tan(x))
Step 1
Use the properties of logarithms to simplify the differentiation.
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Step 1.1
Rewrite as .
Step 1.2
Expand by moving outside the logarithm.
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate using the Product Rule which states that is where and .
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
The derivative of with respect to is .
Step 4.3
Replace all occurrences of with .
Step 5
Convert from to .
Step 6
The derivative of with respect to is .
Step 7
The derivative of with respect to is .
Step 8
Simplify.
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Step 8.1
Apply the distributive property.
Step 8.2
Remove parentheses.
Step 8.3
Reorder terms.
Step 8.4
Simplify each term.
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Step 8.4.1
Rewrite in terms of sines and cosines.
Step 8.4.2
Combine and .
Step 8.4.3
Rewrite in terms of sines and cosines.
Step 8.4.4
Multiply .
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Step 8.4.4.1
Combine and .
Step 8.4.4.2
Combine and .
Step 8.4.5
Rewrite in terms of sines and cosines.
Step 8.4.6
Combine.
Step 8.4.7
Cancel the common factor of .
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Step 8.4.7.1
Cancel the common factor.
Step 8.4.7.2
Rewrite the expression.
Step 8.4.8
Cancel the common factor of .
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Step 8.4.8.1
Cancel the common factor.
Step 8.4.8.2
Divide by .
Step 8.4.9
Rewrite in terms of sines and cosines.
Step 8.4.10
Combine and .
Step 8.4.11
Rewrite in terms of sines and cosines.
Step 8.4.12
Apply the product rule to .
Step 8.4.13
One to any power is one.
Step 8.4.14
Combine and .
Step 8.4.15
Combine and .
Step 8.5
Simplify each term.
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Step 8.5.1
Separate fractions.
Step 8.5.2
Convert from to .
Step 8.5.3
Divide by .
Step 8.5.4
Simplify the numerator.
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Step 8.5.4.1
Separate fractions.
Step 8.5.4.2
Convert from to .
Step 8.5.4.3
Divide by .