Calculus Examples

Find the Derivative - d/dx y=sin(9x)^( natural log of x)
Step 1
Use the properties of logarithms to simplify the differentiation.
Tap for more steps...
Step 1.1
Rewrite as .
Step 1.2
Expand by moving outside the logarithm.
Step 2
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate using the Product Rule which states that is where and .
Step 4
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 4.1
To apply the Chain Rule, set as .
Step 4.2
The derivative of with respect to is .
Step 4.3
Replace all occurrences of with .
Step 5
Convert from to .
Step 6
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 6.1
To apply the Chain Rule, set as .
Step 6.2
The derivative of with respect to is .
Step 6.3
Replace all occurrences of with .
Step 7
Differentiate.
Tap for more steps...
Step 7.1
Since is constant with respect to , the derivative of with respect to is .
Step 7.2
Differentiate using the Power Rule which states that is where .
Step 7.3
Simplify the expression.
Tap for more steps...
Step 7.3.1
Multiply by .
Step 7.3.2
Move to the left of .
Step 8
The derivative of with respect to is .
Step 9
Combine and .
Step 10
Simplify.
Tap for more steps...
Step 10.1
Apply the distributive property.
Step 10.2
Combine terms.
Tap for more steps...
Step 10.2.1
Move to the left of .
Step 10.2.2
Combine and .
Step 10.3
Reorder terms.