Calculus Examples

Find the Derivative - d/dx y=arctan( square root of (1+x)/(1-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
Multiply the exponents in .
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Step 3.1
Apply the power rule and multiply exponents, .
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 4
Simplify.
Step 5
Write as a fraction with a common denominator.
Step 6
Combine the numerators over the common denominator.
Step 7
Add and .
Step 8
Add and .
Step 9
Add and .
Step 10
Multiply by the reciprocal of the fraction to divide by .
Step 11
Multiply by .
Step 12
Differentiate using the chain rule, which states that is where and .
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Step 12.1
To apply the Chain Rule, set as .
Step 12.2
Differentiate using the Power Rule which states that is where .
Step 12.3
Replace all occurrences of with .
Step 13
To write as a fraction with a common denominator, multiply by .
Step 14
Combine and .
Step 15
Combine the numerators over the common denominator.
Step 16
Simplify the numerator.
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Step 16.1
Multiply by .
Step 16.2
Subtract from .
Step 17
Combine fractions.
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Step 17.1
Move the negative in front of the fraction.
Step 17.2
Multiply by .
Step 17.3
Multiply by .
Step 18
Differentiate using the Quotient Rule which states that is where and .
Step 19
Differentiate.
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Step 19.1
By the Sum Rule, the derivative of with respect to is .
Step 19.2
Since is constant with respect to , the derivative of with respect to is .
Step 19.3
Add and .
Step 19.4
Differentiate using the Power Rule which states that is where .
Step 19.5
Multiply by .
Step 19.6
By the Sum Rule, the derivative of with respect to is .
Step 19.7
Since is constant with respect to , the derivative of with respect to is .
Step 19.8
Add and .
Step 19.9
Since is constant with respect to , the derivative of with respect to is .
Step 19.10
Multiply.
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Step 19.10.1
Multiply by .
Step 19.10.2
Multiply by .
Step 19.11
Differentiate using the Power Rule which states that is where .
Step 19.12
Simplify terms.
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Step 19.12.1
Multiply by .
Step 19.12.2
Add and .
Step 19.12.3
Add and .
Step 19.12.4
Add and .
Step 19.12.5
Multiply by .
Step 19.12.6
Move to the left of .
Step 20
Cancel the common factors.
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Step 20.1
Factor out of .
Step 20.2
Cancel the common factor.
Step 20.3
Rewrite the expression.
Step 21
Cancel the common factor of and .
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Step 21.1
Multiply by .
Step 21.2
Cancel the common factors.
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Step 21.2.1
Factor out of .
Step 21.2.2
Cancel the common factor.
Step 21.2.3
Rewrite the expression.
Step 22
Simplify.
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Step 22.1
Change the sign of the exponent by rewriting the base as its reciprocal.
Step 22.2
Apply the product rule to .
Step 22.3
Apply the distributive property.
Step 22.4
Combine terms.
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Step 22.4.1
Multiply by .
Step 22.4.2
Multiply by .
Step 22.4.3
Multiply by .
Step 22.5
Reorder terms.
Step 22.6
Factor out of .
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Step 22.6.1
Factor out of .
Step 22.6.2
Factor out of .
Step 22.6.3
Factor out of .
Step 22.7
Reorder terms.
Step 22.8
Factor out of .
Step 22.9
Cancel the common factors.
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Step 22.9.1
Factor out of .
Step 22.9.2
Cancel the common factor.
Step 22.9.3
Rewrite the expression.
Step 22.10
Move to the denominator using the negative exponent rule .
Step 22.11
Move to the left of .