Calculus Examples

Find the Derivative - d/dx 9x^( natural log of x)
Step 1
Since is constant with respect to , the derivative of with respect to is .
Step 2
Use the properties of logarithms to simplify the differentiation.
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Step 2.1
Rewrite as .
Step 2.2
Expand by moving outside the logarithm.
Step 3
Raise to the power of .
Step 4
Raise to the power of .
Step 5
Use the power rule to combine exponents.
Step 6
Add and .
Step 7
Differentiate using the chain rule, which states that is where and .
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Step 7.1
To apply the Chain Rule, set as .
Step 7.2
Differentiate using the Exponential Rule which states that is where =.
Step 7.3
Replace all occurrences of with .
Step 8
Differentiate using the chain rule, which states that is where and .
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Step 8.1
To apply the Chain Rule, set as .
Step 8.2
Differentiate using the Power Rule which states that is where .
Step 8.3
Replace all occurrences of with .
Step 9
Multiply by .
Step 10
The derivative of with respect to is .
Step 11
Combine fractions.
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Step 11.1
Combine and .
Step 11.2
Combine and .
Step 11.3
Combine and .
Step 11.4
Move to the left of .
Step 12
Simplify the numerator.
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Step 12.1
Simplify by moving inside the logarithm.
Step 12.2
Reorder factors in .