Calculus Examples

Use Logarithmic Differentiation to Find the Derivative
Step 1
Let , take the natural logarithm of both sides .
Step 2
Expand the right hand side.
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Use to rewrite as .
Expand by moving outside the logarithm.
Step 3
Differentiate the expression using the chain rule, keeping in mind that is a function of .
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Differentiate the left hand side using the chain rule.
Differentiate the right hand side.
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Differentiate .
Differentiate using the Product Rule which states that is where and .
The derivative of with respect to is .
Combine fractions.
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Combine and .
Move to the denominator using the negative exponent rule .
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Subtract from .
Differentiate using the Power Rule which states that is where .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Combine and .
Move to the denominator using the negative exponent rule .
Step 4
Isolate and substitute the original function for in the right hand side.
Step 5
Simplify the right hand side.
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Apply the distributive property.
Combine and .
Combine and .
Simplify each term.
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Factor out of .
Cancel the common factors.
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Multiply by .
Cancel the common factor.
Rewrite the expression.
Divide by .
Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify the numerator.
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To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Move to the left of .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Use to rewrite as .
Use to rewrite as .
Move to the left of .
Simplify each term.
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To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Move to the left of .
Factor out of .
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Reorder and .
Factor out of .
Factor out of .
Factor out of .
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