# Find the Derivative – d/dx y=x^( natural log of 4x) Use the properties of logarithms to simplify the differentiation.
Rewrite as .
Expand by moving outside the logarithm.
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Exponential Rule which states that is where =.
Replace all occurrences of with .
Differentiate using the Product Rule which states that is where and .
The derivative of with respect to is .
Combine and .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
Differentiate.
Combine and .
Since is constant with respect to , the derivative of with respect to is .
Simplify terms.
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Differentiate using the Power Rule which states that is where .
Multiply by .
Simplify.
Apply the distributive property.
Combine terms.
Combine and .
Combine and .
Find the Derivative – d/dx y=x^( natural log of 4x)     