Calculus Examples

Find the Derivative - d/dx y=( log base 6 of x)/(3+x^6)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
The derivative of with respect to is .
Step 3
Differentiate.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Add and .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Multiply by .
Step 4
Simplify.
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Step 4.1
Apply the distributive property.
Step 4.2
Simplify the numerator.
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Step 4.2.1
Simplify each term.
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Step 4.2.1.1
Combine and .
Step 4.2.1.2
Cancel the common factor of .
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Step 4.2.1.2.1
Factor out of .
Step 4.2.1.2.2
Factor out of .
Step 4.2.1.2.3
Cancel the common factor.
Step 4.2.1.2.4
Rewrite the expression.
Step 4.2.1.3
Combine and .
Step 4.2.1.4
Simplify by moving inside the logarithm.
Step 4.2.2
Reorder factors in .
Step 4.3
Combine terms.
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Step 4.3.1
Multiply by .
Step 4.3.2
Combine.
Step 4.3.3
Apply the distributive property.
Step 4.3.4
Cancel the common factor of .
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Step 4.3.4.1
Cancel the common factor.
Step 4.3.4.2
Rewrite the expression.
Step 4.3.5
Cancel the common factor of .
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Step 4.3.5.1
Factor out of .
Step 4.3.5.2
Cancel the common factor.
Step 4.3.5.3
Rewrite the expression.
Step 4.3.6
Multiply by by adding the exponents.
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Step 4.3.6.1
Multiply by .
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Step 4.3.6.1.1
Raise to the power of .
Step 4.3.6.1.2
Use the power rule to combine exponents.
Step 4.3.6.2
Add and .
Step 4.3.7
Raise to the power of .
Step 4.3.8
Use the power rule to combine exponents.
Step 4.3.9
Add and .
Step 4.4
Reorder terms.