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 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
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 9.4
Combine and .
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
Combine fractions.
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Step 13.1
Add and .
Step 13.2
Combine and .
Step 13.3
Combine and .
Step 14
Raise to the power of .
Step 15
Raise to the power of .
Step 16
Use the power rule to combine exponents.
Step 17
Reduce the expression by cancelling the common factors.
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Step 17.1
Add and .
Step 17.2
Cancel the common factor.
Step 17.3
Rewrite the expression.
Step 18
Differentiate using the Power Rule which states that is where .
Step 19
Multiply by .
Step 20
To write as a fraction with a common denominator, multiply by .
Step 21
Combine the numerators over the common denominator.
Step 22
Multiply by by adding the exponents.
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Step 22.1
Use the power rule to combine exponents.
Step 22.2
Combine the numerators over the common denominator.
Step 22.3
Add and .
Step 22.4
Divide by .
Step 23
Simplify .
Step 24
Add and .
Step 25
Multiply by .
Step 26
Multiply by by adding the exponents.
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Step 26.1
Move .
Step 26.2
Use the power rule to combine exponents.
Step 26.3
Combine the numerators over the common denominator.
Step 26.4
Add and .
Step 26.5
Divide by .
Step 27
Simplify .
Step 28
Simplify.
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Step 28.1
Apply the distributive property.
Step 28.2
Combine terms.
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Step 28.2.1
Multiply by by adding the exponents.
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Step 28.2.1.1
Multiply by .
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Step 28.2.1.1.1
Raise to the power of .
Step 28.2.1.1.2
Use the power rule to combine exponents.
Step 28.2.1.2
Add and .
Step 28.2.2
Move to the left of .
Step 28.2.3
Rewrite as .