# Calculus Examples

Step 1

Let , take the natural logarithm of both sides .

Step 2

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

Add and .

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 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

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 .