Calculus Examples

Find the Derivative - d/dx f(x)=(6x)^( natural log of 6x)
Step 1
Simplify with factoring out.
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Step 1.1
Factor out of .
Step 1.2
Apply the product rule to .
Step 2
Differentiate using the Product Rule which states that is where and .
Step 3
Use the properties of logarithms to simplify the differentiation.
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Step 3.1
Rewrite as .
Step 3.2
Expand by moving outside the logarithm.
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate using the Product Rule which states that is where and .
Step 6
The derivative of with respect to is .
Step 7
Combine and .
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
The derivative of with respect to is .
Step 8.3
Replace all occurrences of with .
Step 9
Differentiate.
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Step 9.1
Combine and .
Step 9.2
Since is constant with respect to , the derivative of with respect to is .
Step 9.3
Simplify terms.
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Step 9.3.1
Combine and .
Step 9.3.2
Cancel the common factor of .
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Step 9.3.2.1
Cancel the common factor.
Step 9.3.2.2
Rewrite the expression.
Step 9.4
Differentiate using the Power Rule which states that is where .
Step 9.5
Multiply by .
Step 10
Differentiate using the chain rule, which states that is where and .
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Step 10.1
To apply the Chain Rule, set as .
Step 10.2
Differentiate using the Exponential Rule which states that is where =.
Step 10.3
Replace all occurrences of with .
Step 11
Differentiate using the chain rule, which states that is where and .
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Step 11.1
To apply the Chain Rule, set as .
Step 11.2
The derivative of with respect to is .
Step 11.3
Replace all occurrences of with .
Step 12
Differentiate using the Constant Multiple Rule.
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Step 12.1
Combine and .
Step 12.2
Simplify terms.
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Step 12.2.1
Combine and .
Step 12.2.2
Cancel the common factor of and .
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Step 12.2.2.1
Factor out of .
Step 12.2.2.2
Cancel the common factors.
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Step 12.2.2.2.1
Factor out of .
Step 12.2.2.2.2
Cancel the common factor.
Step 12.2.2.2.3
Rewrite the expression.
Step 12.2.3
Combine and .
Step 12.2.4
Cancel the common factor of and .
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Step 12.2.4.1
Factor out of .
Step 12.2.4.2
Cancel the common factors.
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Step 12.2.4.2.1
Raise to the power of .
Step 12.2.4.2.2
Factor out of .
Step 12.2.4.2.3
Cancel the common factor.
Step 12.2.4.2.4
Rewrite the expression.
Step 12.2.4.2.5
Divide by .
Step 12.3
Since is constant with respect to , the derivative of with respect to is .
Step 13
Raise to the power of .
Step 14
Use the power rule to combine exponents.
Step 15
Simplify by subtracting numbers.
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Step 15.1
Subtract from .
Step 15.2
Add and .
Step 16
Differentiate using the Power Rule which states that is where .
Step 17
Multiply by .
Step 18
Simplify.
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Step 18.1
Apply the distributive property.
Step 18.2
Apply the distributive property.
Step 18.3
Combine terms.
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Step 18.3.1
Combine and .
Step 18.3.2
Combine and .
Step 18.3.3
Combine and .
Step 18.3.4
Combine and .
Step 18.3.5
To write as a fraction with a common denominator, multiply by .
Step 18.3.6
Combine the numerators over the common denominator.
Step 18.3.7
Raise to the power of .
Step 18.3.8
Use the power rule to combine exponents.
Step 18.3.9
Subtract from .
Step 18.3.10
Add and .
Step 18.3.11
Combine the numerators over the common denominator.
Step 18.4
Reorder terms.
Step 18.5
Factor out of .
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Step 18.5.1
Factor out of .
Step 18.5.2
Factor out of .
Step 18.5.3
Factor out of .
Step 18.5.4
Factor out of .
Step 18.5.5
Factor out of .