Calculus Examples

Find the Derivative - d/dx y=(8 natural log of x+5)/(x^2)
Step 1
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Multiply the exponents in .
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Step 3.1
Apply the power rule and multiply exponents, .
Step 3.2
Multiply by .
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
The derivative of with respect to is .
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate.
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Step 5.1
Combine and .
Step 5.2
By the Sum Rule, the derivative of with respect to is .
Step 5.3
Differentiate using the Power Rule which states that is where .
Step 5.4
Since is constant with respect to , the derivative of with respect to is .
Step 5.5
Simplify the expression.
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Step 5.5.1
Add and .
Step 5.5.2
Multiply by .
Step 6
Multiply by .
Step 7
Simplify terms.
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Step 7.1
Combine.
Step 7.2
Apply the distributive property.
Step 7.3
Cancel the common factor of .
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Step 7.3.1
Cancel the common factor.
Step 7.3.2
Rewrite the expression.
Step 8
Differentiate using the Power Rule which states that is where .
Step 9
Combine fractions.
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Step 9.1
Multiply by .
Step 9.2
Combine and .
Step 10
Simplify.
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Step 10.1
Apply the distributive property.
Step 10.2
Apply the distributive property.
Step 10.3
Simplify the numerator.
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Step 10.3.1
Simplify each term.
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Step 10.3.1.1
Simplify by moving inside the logarithm.
Step 10.3.1.2
Apply the distributive property.
Step 10.3.1.3
Rewrite using the commutative property of multiplication.
Step 10.3.1.4
Multiply .
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Step 10.3.1.4.1
Multiply by .
Step 10.3.1.4.2
Reorder and .
Step 10.3.1.4.3
Simplify by moving inside the logarithm.
Step 10.3.1.5
Simplify each term.
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Step 10.3.1.5.1
Multiply by by adding the exponents.
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Step 10.3.1.5.1.1
Move .
Step 10.3.1.5.1.2
Multiply by .
Step 10.3.1.5.2
Multiply the exponents in .
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Step 10.3.1.5.2.1
Apply the power rule and multiply exponents, .
Step 10.3.1.5.2.2
Multiply by .
Step 10.3.1.6
Apply the distributive property.
Step 10.3.1.7
Multiply .
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Step 10.3.1.7.1
Multiply by .
Step 10.3.1.7.2
Simplify by moving inside the logarithm.
Step 10.3.1.8
Multiply .
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Step 10.3.1.8.1
Multiply by .
Step 10.3.1.8.2
Reorder and .
Step 10.3.1.8.3
Simplify by moving inside the logarithm.
Step 10.3.1.9
Simplify each term.
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Step 10.3.1.9.1
Rewrite using the commutative property of multiplication.
Step 10.3.1.9.2
Multiply the exponents in .
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Step 10.3.1.9.2.1
Apply the power rule and multiply exponents, .
Step 10.3.1.9.2.2
Multiply by .
Step 10.3.1.9.3
Multiply the exponents in .
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Step 10.3.1.9.3.1
Apply the power rule and multiply exponents, .
Step 10.3.1.9.3.2
Multiply by .
Step 10.3.2
Reorder factors in .
Step 10.4
Multiply by by adding the exponents.
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Step 10.4.1
Multiply by .
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Step 10.4.1.1
Raise to the power of .
Step 10.4.1.2
Use the power rule to combine exponents.
Step 10.4.2
Add and .
Step 10.5
Factor out of .
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Step 10.5.1
Factor out of .
Step 10.5.2
Factor out of .
Step 10.5.3
Factor out of .
Step 10.5.4
Factor out of .
Step 10.5.5
Factor out of .
Step 10.6
Factor out of .
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Step 10.6.1
Factor out of .
Step 10.6.2
Factor out of .
Step 10.6.3
Factor out of .
Step 10.7
Expand by moving outside the logarithm.
Step 10.8
Expand by moving outside the logarithm.
Step 10.9
Cancel the common factors.
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Step 10.9.1
Factor out of .
Step 10.9.2
Cancel the common factor.
Step 10.9.3
Rewrite the expression.
Step 10.10
Simplify the numerator.
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Step 10.10.1
Multiply by .
Step 10.10.2
Multiply by .
Step 10.10.3
Factor out of .
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Step 10.10.3.1
Factor out of .
Step 10.10.3.2
Factor out of .
Step 10.10.3.3
Factor out of .
Step 10.10.3.4
Factor out of .
Step 10.10.3.5
Factor out of .