Calculus Examples

Find the 2nd Derivative f(x)=( natural log of x)/(11x)
Step 1
Find the first derivative.
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Step 1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Differentiate using the Quotient Rule which states that is where and .
Step 1.3
The derivative of with respect to is .
Step 1.4
Differentiate using the Power Rule.
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Step 1.4.1
Combine and .
Step 1.4.2
Cancel the common factor of .
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Step 1.4.2.1
Cancel the common factor.
Step 1.4.2.2
Rewrite the expression.
Step 1.4.3
Differentiate using the Power Rule which states that is where .
Step 1.4.4
Combine fractions.
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Step 1.4.4.1
Multiply by .
Step 1.4.4.2
Multiply by .
Step 2
Find the second derivative.
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Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Differentiate.
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Step 2.3.1
Multiply the exponents in .
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Step 2.3.1.1
Apply the power rule and multiply exponents, .
Step 2.3.1.2
Multiply by .
Step 2.3.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.4
Add and .
Step 2.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
The derivative of with respect to is .
Step 2.5
Differentiate using the Power Rule.
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Step 2.5.1
Combine and .
Step 2.5.2
Cancel the common factor of and .
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Step 2.5.2.1
Factor out of .
Step 2.5.2.2
Cancel the common factors.
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Step 2.5.2.2.1
Raise to the power of .
Step 2.5.2.2.2
Factor out of .
Step 2.5.2.2.3
Cancel the common factor.
Step 2.5.2.2.4
Rewrite the expression.
Step 2.5.2.2.5
Divide by .
Step 2.5.3
Differentiate using the Power Rule which states that is where .
Step 2.5.4
Simplify with factoring out.
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Step 2.5.4.1
Multiply by .
Step 2.5.4.2
Factor out of .
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Step 2.5.4.2.1
Factor out of .
Step 2.5.4.2.2
Factor out of .
Step 2.5.4.2.3
Factor out of .
Step 2.6
Cancel the common factors.
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Step 2.6.1
Factor out of .
Step 2.6.2
Cancel the common factor.
Step 2.6.3
Rewrite the expression.
Step 2.7
Multiply by .
Step 2.8
Simplify.
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Step 2.8.1
Apply the distributive property.
Step 2.8.2
Simplify the numerator.
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Step 2.8.2.1
Simplify each term.
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Step 2.8.2.1.1
Multiply by .
Step 2.8.2.1.2
Multiply .
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Step 2.8.2.1.2.1
Multiply by .
Step 2.8.2.1.2.2
Simplify by moving inside the logarithm.
Step 2.8.2.2
Subtract from .
Step 2.8.3
Rewrite as .
Step 2.8.4
Factor out of .
Step 2.8.5
Factor out of .
Step 2.8.6
Move the negative in front of the fraction.
Step 3
Find the third derivative.
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Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.3
Differentiate.
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Step 3.3.1
Multiply the exponents in .
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Step 3.3.1.1
Apply the power rule and multiply exponents, .
Step 3.3.1.2
Multiply by .
Step 3.3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.4
Add and .
Step 3.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Differentiate using the chain rule, which states that is where and .
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Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
The derivative of with respect to is .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Differentiate using the Power Rule.
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Step 3.5.1
Combine and .
Step 3.5.2
Cancel the common factor of and .
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Step 3.5.2.1
Factor out of .
Step 3.5.2.2
Cancel the common factors.
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Step 3.5.2.2.1
Multiply by .
Step 3.5.2.2.2
Cancel the common factor.
Step 3.5.2.2.3
Rewrite the expression.
Step 3.5.2.2.4
Divide by .
Step 3.5.3
Differentiate using the Power Rule which states that is where .
Step 3.5.4
Multiply by .
Step 3.6
Raise to the power of .
Step 3.7
Raise to the power of .
Step 3.8
Use the power rule to combine exponents.
Step 3.9
Add and .
Step 3.10
Differentiate using the Power Rule which states that is where .
Step 3.11
Simplify with factoring out.
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Step 3.11.1
Multiply by .
Step 3.11.2
Factor out of .
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Step 3.11.2.1
Factor out of .
Step 3.11.2.2
Factor out of .
Step 3.11.2.3
Factor out of .
Step 3.12
Cancel the common factors.
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Step 3.12.1
Factor out of .
Step 3.12.2
Cancel the common factor.
Step 3.12.3
Rewrite the expression.
Step 3.13
Multiply by .
Step 3.14
Move to the left of .
Step 3.15
Simplify.
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Step 3.15.1
Apply the distributive property.
Step 3.15.2
Simplify the numerator.
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Step 3.15.2.1
Simplify each term.
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Step 3.15.2.1.1
Multiply by .
Step 3.15.2.1.2
Multiply .
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Step 3.15.2.1.2.1
Multiply by .
Step 3.15.2.1.2.2
Simplify by moving inside the logarithm.
Step 3.15.2.1.3
Multiply the exponents in .
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Step 3.15.2.1.3.1
Apply the power rule and multiply exponents, .
Step 3.15.2.1.3.2
Multiply by .
Step 3.15.2.2
Subtract from .
Step 3.15.3
Rewrite as .
Step 3.15.4
Factor out of .
Step 3.15.5
Factor out of .
Step 3.15.6
Move the negative in front of the fraction.
Step 3.15.7
Multiply by .
Step 3.15.8
Multiply by .
Step 4
Find the fourth derivative.
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Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Quotient Rule which states that is where and .
Step 4.3
Differentiate.
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Step 4.3.1
Multiply the exponents in .
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Step 4.3.1.1
Apply the power rule and multiply exponents, .
Step 4.3.1.2
Multiply by .
Step 4.3.2
By the Sum Rule, the derivative of with respect to is .
Step 4.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.4
Add and .
Step 4.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.4
Differentiate using the chain rule, which states that is where and .
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Step 4.4.1
To apply the Chain Rule, set as .
Step 4.4.2
The derivative of with respect to is .
Step 4.4.3
Replace all occurrences of with .
Step 4.5
Differentiate using the Power Rule.
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Step 4.5.1
Combine and .
Step 4.5.2
Cancel the common factor of and .
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Step 4.5.2.1
Multiply by .
Step 4.5.2.2
Cancel the common factors.
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Step 4.5.2.2.1
Factor out of .
Step 4.5.2.2.2
Cancel the common factor.
Step 4.5.2.2.3
Rewrite the expression.
Step 4.5.3
Differentiate using the Power Rule which states that is where .
Step 4.5.4
Simplify terms.
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Step 4.5.4.1
Multiply by .
Step 4.5.4.2
Combine and .
Step 4.5.4.3
Combine and .
Step 4.5.4.4
Cancel the common factor of and .
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Step 4.5.4.4.1
Factor out of .
Step 4.5.4.4.2
Cancel the common factors.
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Step 4.5.4.4.2.1
Multiply by .
Step 4.5.4.4.2.2
Cancel the common factor.
Step 4.5.4.4.2.3
Rewrite the expression.
Step 4.5.4.4.2.4
Divide by .
Step 4.5.5
Differentiate using the Power Rule which states that is where .
Step 4.5.6
Combine fractions.
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Step 4.5.6.1
Multiply by .
Step 4.5.6.2
Multiply by .
Step 4.6
Simplify.
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Step 4.6.1
Apply the distributive property.
Step 4.6.2
Apply the distributive property.
Step 4.6.3
Simplify the numerator.
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Step 4.6.3.1
Simplify each term.
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Step 4.6.3.1.1
Multiply by .
Step 4.6.3.1.2
Multiply .
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Step 4.6.3.1.2.1
Multiply by .
Step 4.6.3.1.2.2
Simplify by moving inside the logarithm.
Step 4.6.3.1.3
Multiply the exponents in .
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Step 4.6.3.1.3.1
Apply the power rule and multiply exponents, .
Step 4.6.3.1.3.2
Multiply by .
Step 4.6.3.2
Subtract from .
Step 4.6.3.3
Reorder factors in .
Step 4.6.4
Factor out of .
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Step 4.6.4.1
Factor out of .
Step 4.6.4.2
Factor out of .
Step 4.6.4.3
Factor out of .
Step 4.6.5
Expand by moving outside the logarithm.
Step 4.6.6
Cancel the common factors.
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Step 4.6.6.1
Factor out of .
Step 4.6.6.2
Cancel the common factor.
Step 4.6.6.3
Rewrite the expression.
Step 4.6.7
Factor out of .
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Step 4.6.7.1
Factor out of .
Step 4.6.7.2
Factor out of .
Step 4.6.7.3
Factor out of .
Step 4.6.8
Rewrite as .
Step 4.6.9
Factor out of .
Step 4.6.10
Factor out of .
Step 4.6.11
Move the negative in front of the fraction.
Step 5
The fourth derivative of with respect to is .