# Calculus Examples

Step 1

Let , take the natural logarithm of both sides .

Step 2

Expand by moving outside the logarithm.

Step 3

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

Add and .

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

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

Add and .

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 .