Calculus Examples

Find the Derivative - d/dx y = log base 5 of 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 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
Combine and .
Step 10
Move to the denominator using the negative exponent rule .
Step 11
Multiply by .
Step 12
Use the power rule to combine exponents.
Step 13
Simplify the expression.
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Step 13.1
Combine the numerators over the common denominator.
Step 13.2
Add and .
Step 14
Cancel the common factor of .
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Step 14.1
Cancel the common factor.
Step 14.2
Rewrite the expression.
Step 15
Simplify.
Step 16
By the Sum Rule, the derivative of with respect to is .
Step 17
Differentiate using the Power Rule which states that is where .
Step 18
Since is constant with respect to , the derivative of with respect to is .
Step 19
Simplify terms.
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Step 19.1
Add and .
Step 19.2
Combine and .
Step 19.3
Combine and .
Step 19.4
Cancel the common factor of .
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Step 19.4.1
Cancel the common factor.
Step 19.4.2
Rewrite the expression.
Step 20
Simplify.
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Step 20.1
Apply the distributive property.
Step 20.2
Rewrite as .