Calculus Examples

Find the Derivative - d/dx natural log of x- square root of x^2-1
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.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
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
Differentiate using the Power Rule which states that is where .
Step 4.3
Replace all occurrences of with .
Step 5
To write as a fraction with a common denominator, multiply by .
Step 6
Combine and .
Step 7
Combine the numerators over the common denominator.
Step 8
Simplify the numerator.
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Step 8.1
Multiply by .
Step 8.2
Subtract from .
Step 9
Combine fractions.
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Step 9.1
Move the negative in front of the fraction.
Step 9.2
Combine and .
Step 9.3
Move to the denominator using the negative exponent rule .
Step 10
By the Sum Rule, the derivative of with respect to is .
Step 11
Differentiate using the Power Rule which states that is where .
Step 12
Since is constant with respect to , the derivative of with respect to is .
Step 13
Simplify terms.
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Step 13.1
Add and .
Step 13.2
Multiply by .
Step 13.3
Combine and .
Step 13.4
Combine and .
Step 13.5
Factor out of .
Step 14
Cancel the common factors.
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Step 14.1
Factor out of .
Step 14.2
Cancel the common factor.
Step 14.3
Rewrite the expression.
Step 15
Move the negative in front of the fraction.
Step 16
Simplify.
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Step 16.1
Reorder the factors of .
Step 16.2
Multiply by .
Step 16.3
Multiply the numerator and denominator of the fraction by .
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Step 16.3.1
Multiply by .
Step 16.3.2
Combine.
Step 16.4
Apply the distributive property.
Step 16.5
Cancel the common factor of .
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Step 16.5.1
Move the leading negative in into the numerator.
Step 16.5.2
Cancel the common factor.
Step 16.5.3
Rewrite the expression.
Step 16.6
Multiply by .
Step 16.7
Factor out of .
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Step 16.7.1
Reorder the expression.
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Step 16.7.1.1
Reorder and .
Step 16.7.1.2
Reorder and .
Step 16.7.2
Factor out of .
Step 16.7.3
Factor out of .
Step 16.7.4
Factor out of .
Step 16.8
Factor out of .
Step 16.9
Factor out of .
Step 16.10
Factor out of .
Step 16.11
Reorder terms.
Step 16.12
Cancel the common factor.
Step 16.13
Rewrite the expression.
Step 16.14
Move the negative in front of the fraction.