# 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.
Step 2.1
Expand by moving outside the logarithm.
Step 2.2
Raise to the power of .
Step 2.3
Raise to the power of .
Step 2.4
Use the power rule to combine exponents.
Step 2.5
Step 3
Differentiate the expression using the chain rule, keeping in mind that is a function of .
Step 3.1
Differentiate the left hand side using the chain rule.
Step 3.2
Differentiate the right hand side.
Step 3.2.1
Differentiate .
Step 3.2.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.2.1
To apply the Chain Rule, set as .
Step 3.2.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.2.3
Replace all occurrences of with .
Step 3.2.3
The derivative of with respect to is .
Step 3.2.4
Combine fractions.
Step 3.2.4.1
Combine and .
Step 3.2.4.2
Combine and .
Step 3.2.5
Simplify by moving inside the logarithm.
Step 4
Isolate and substitute the original function for in the right hand side.
Step 5
Simplify the right hand side.
Step 5.1
Combine and .
Step 5.2
Cancel the common factor of and .
Step 5.2.1
Factor out of .
Step 5.2.2
Cancel the common factors.
Step 5.2.2.1
Raise to the power of .
Step 5.2.2.2
Factor out of .
Step 5.2.2.3
Cancel the common factor.
Step 5.2.2.4
Rewrite the expression.
Step 5.2.2.5
Divide by .
Step 5.3
Reorder factors in .