Calculus Examples

Find the Derivative - d/dx natural log of square root of (1+sin(x))/(1-sin(x))
Step 1
Use to rewrite as .
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
The derivative of with respect to is .
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate using the chain rule, which states that is where and .
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Step 3.1
To apply the Chain Rule, set as .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Replace all occurrences of with .
Step 4
To write as a fraction with a common denominator, multiply by .
Step 5
Combine and .
Step 6
Combine the numerators over the common denominator.
Step 7
Simplify the numerator.
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Step 7.1
Multiply by .
Step 7.2
Subtract from .
Step 8
Move the negative in front of the fraction.
Step 9
Differentiate using the Quotient Rule which states that is where and .
Step 10
Differentiate.
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Step 10.1
By the Sum Rule, the derivative of with respect to is .
Step 10.2
Since is constant with respect to , the derivative of with respect to is .
Step 10.3
Add and .
Step 11
The derivative of with respect to is .
Step 12
Differentiate.
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Step 12.1
By the Sum Rule, the derivative of with respect to is .
Step 12.2
Since is constant with respect to , the derivative of with respect to is .
Step 12.3
Add and .
Step 12.4
Since is constant with respect to , the derivative of with respect to is .
Step 12.5
Multiply.
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Step 12.5.1
Multiply by .
Step 12.5.2
Multiply by .
Step 13
The derivative of with respect to is .
Step 14
Combine fractions.
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Step 14.1
Multiply by .
Step 14.2
Move to the left of .
Step 15
Simplify.
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Step 15.1
Change the sign of the exponent by rewriting the base as its reciprocal.
Step 15.2
Apply the product rule to .
Step 15.3
Apply the product rule to .
Step 15.4
Apply the distributive property.
Step 15.5
Apply the distributive property.
Step 15.6
Combine terms.
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Step 15.6.1
Multiply by the reciprocal of the fraction to divide by .
Step 15.6.2
Multiply by .
Step 15.6.3
Multiply by .
Step 15.6.4
Multiply by .
Step 15.6.5
Add and .
Step 15.6.6
Add and .
Step 15.6.7
Add and .
Step 15.6.8
Cancel the common factor of .
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Step 15.6.8.1
Cancel the common factor.
Step 15.6.8.2
Rewrite the expression.
Step 15.6.9
Multiply by .
Step 15.6.10
Move to the denominator using the negative exponent rule .
Step 15.6.11
Multiply by by adding the exponents.
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Step 15.6.11.1
Move .
Step 15.6.11.2
Use the power rule to combine exponents.
Step 15.6.11.3
To write as a fraction with a common denominator, multiply by .
Step 15.6.11.4
Combine and .
Step 15.6.11.5
Combine the numerators over the common denominator.
Step 15.6.11.6
Simplify the numerator.
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Step 15.6.11.6.1
Multiply by .
Step 15.6.11.6.2
Add and .
Step 15.6.12
Multiply by .
Step 15.6.13
Multiply by by adding the exponents.
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Step 15.6.13.1
Move .
Step 15.6.13.2
Use the power rule to combine exponents.
Step 15.6.13.3
Combine the numerators over the common denominator.
Step 15.6.13.4
Add and .
Step 15.6.13.5
Divide by .
Step 15.6.14
Simplify .
Step 15.6.15
Move to the denominator using the negative exponent rule .
Step 15.6.16
Simplify the denominator.
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Step 15.6.16.1
Multiply by by adding the exponents.
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Step 15.6.16.1.1
Move .
Step 15.6.16.1.2
Use the power rule to combine exponents.
Step 15.6.16.1.3
Combine the numerators over the common denominator.
Step 15.6.16.1.4
Add and .
Step 15.6.16.1.5
Divide by .
Step 15.6.16.2
Simplify .