# Calculus Examples

Use Logarithmic Differentiation to Find the Derivative
Step 1
Let , take the natural logarithm of both sides .
Step 2
Expand by moving outside the logarithm.
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 Product Rule which states that is where and .
Step 3.2.3
Differentiate using the chain rule, which states that is where and .
Step 3.2.3.1
To apply the Chain Rule, set as .
Step 3.2.3.2
The derivative of with respect to is .
Step 3.2.3.3
Replace all occurrences of with .
Step 3.2.4
Convert from to .
Step 3.2.5
The derivative of with respect to is .
Step 3.2.6
Raise to the power of .
Step 3.2.7
Raise to the power of .
Step 3.2.8
Use the power rule to combine exponents.
Step 3.2.9
Step 3.2.10
The derivative of with respect to is .
Step 3.2.11
Simplify.
Step 3.2.11.1
Reorder terms.
Step 3.2.11.2
Simplify each term.
Step 3.2.11.2.1
Rewrite in terms of sines and cosines.
Step 3.2.11.2.2
Combine and .
Step 3.2.11.3
Simplify each term.
Step 3.2.11.3.1
Factor out of .
Step 3.2.11.3.2
Separate fractions.
Step 3.2.11.3.3
Convert from to .
Step 3.2.11.3.4
Divide by .
Step 4
Isolate and substitute the original function for in the right hand side.
Step 5
Simplify the right hand side.
Step 5.1
Simplify each term.
Step 5.1.1
Rewrite in terms of sines and cosines.
Step 5.1.2
Multiply .
Step 5.1.2.1
Combine and .
Step 5.1.2.2
Raise to the power of .
Step 5.1.2.3
Raise to the power of .
Step 5.1.2.4
Use the power rule to combine exponents.
Step 5.1.2.5
Step 5.2
Apply the distributive property.
Step 5.3
Combine and .
Step 5.4
Multiply by by adding the exponents.
Step 5.4.1
Move .
Step 5.4.2
Multiply by .
Step 5.4.2.1
Raise to the power of .
Step 5.4.2.2
Use the power rule to combine exponents.
Step 5.5
Cancel the common factor of and .
Step 5.5.1
Factor out of .
Step 5.5.2
Cancel the common factors.
Step 5.5.2.1
Multiply by .
Step 5.5.2.2
Cancel the common factor.
Step 5.5.2.3
Rewrite the expression.
Step 5.5.2.4
Divide by .
Step 5.6
Reorder factors in .