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Calculus Examples
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
The derivative of with respect to is .
Step 4
Step 4.1
Use the properties of logarithms to simplify the differentiation.
Step 4.1.1
Rewrite as .
Step 4.1.2
Expand by moving outside the logarithm.
Step 4.2
Differentiate using the chain rule, which states that is where and .
Step 4.2.1
To apply the Chain Rule, set as .
Step 4.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.2.3
Replace all occurrences of with .
Step 4.3
Differentiate using the Product Rule which states that is where and .
Step 4.4
The derivative of with respect to is .
Step 4.5
Combine fractions.
Step 4.5.1
Combine and .
Step 4.5.2
Move to the denominator using the negative exponent rule .
Step 4.6
Multiply by by adding the exponents.
Step 4.6.1
Multiply by .
Step 4.6.1.1
Raise to the power of .
Step 4.6.1.2
Use the power rule to combine exponents.
Step 4.6.2
Write as a fraction with a common denominator.
Step 4.6.3
Combine the numerators over the common denominator.
Step 4.6.4
Subtract from .
Step 4.7
Differentiate using the Power Rule which states that is where .
Step 4.8
To write as a fraction with a common denominator, multiply by .
Step 4.9
Combine and .
Step 4.10
Combine the numerators over the common denominator.
Step 4.11
Simplify the numerator.
Step 4.11.1
Multiply by .
Step 4.11.2
Subtract from .
Step 4.12
Move the negative in front of the fraction.
Step 4.13
Combine and .
Step 4.14
Combine and .
Step 4.15
Move to the denominator using the negative exponent rule .
Step 4.16
Simplify.
Step 4.16.1
Apply the distributive property.
Step 4.16.2
Combine terms.
Step 4.16.2.1
Combine and .
Step 4.16.2.2
Combine and .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .