Calculus Examples

Find the Derivative - d/d@VAR f(x) = log base 8 of (x^2-5)/(x-10)
Step 1
Differentiate using the chain rule, which states that is where and .
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Step 1.1
To apply the Chain Rule, set as .
Step 1.2
The derivative of with respect to is .
Step 1.3
Replace all occurrences of with .
Step 2
Combine and .
Step 3
Multiply by the reciprocal of the fraction to divide by .
Step 4
Multiply by .
Step 5
Differentiate using the Quotient Rule which states that is where and .
Step 6
Differentiate.
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Step 6.1
By the Sum Rule, the derivative of with respect to is .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Since is constant with respect to , the derivative of with respect to is .
Step 6.4
Simplify the expression.
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Step 6.4.1
Add and .
Step 6.4.2
Move to the left of .
Step 6.5
By the Sum Rule, the derivative of with respect to is .
Step 6.6
Differentiate using the Power Rule which states that is where .
Step 6.7
Since is constant with respect to , the derivative of with respect to is .
Step 6.8
Combine fractions.
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Step 6.8.1
Add and .
Step 6.8.2
Multiply by .
Step 6.8.3
Multiply by .
Step 7
Cancel the common factors.
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Step 7.1
Factor out of .
Step 7.2
Cancel the common factor.
Step 7.3
Rewrite the expression.
Step 8
Simplify.
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Step 8.1
Apply the distributive property.
Step 8.2
Apply the distributive property.
Step 8.3
Apply the distributive property.
Step 8.4
Apply the distributive property.
Step 8.5
Simplify the numerator.
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Step 8.5.1
Simplify each term.
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Step 8.5.1.1
Multiply by by adding the exponents.
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Step 8.5.1.1.1
Move .
Step 8.5.1.1.2
Multiply by .
Step 8.5.1.2
Multiply by .
Step 8.5.1.3
Multiply by .
Step 8.5.2
Subtract from .
Step 8.6
Reorder terms.