Calculus Examples

Find the Derivative - d/dx ((x^2+24)^(1/2)-x^2(x^2+24)^(-1/2))/(x^2+24)
Step 1
Move the negative in front of the fraction.
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
By the Sum Rule, the derivative of with respect to is .
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 Power Rule which states that is where .
Step 4.3
Replace all occurrences of with .
Step 5
To write as a fraction with a common denominator, multiply by .
Step 6
Combine and .
Step 7
Combine the numerators over the common denominator.
Step 8
Simplify the numerator.
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Step 8.1
Multiply by .
Step 8.2
Subtract from .
Step 9
Differentiate.
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Step 9.1
Move the negative in front of the fraction.
Step 9.2
Combine fractions.
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Step 9.2.1
Combine and .
Step 9.2.2
Move to the denominator using the negative exponent rule .
Step 9.3
By the Sum Rule, the derivative of with respect to is .
Step 9.4
Differentiate using the Power Rule which states that is where .
Step 9.5
Since is constant with respect to , the derivative of with respect to is .
Step 9.6
Simplify terms.
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Step 9.6.1
Add and .
Step 9.6.2
Combine and .
Step 9.6.3
Combine and .
Step 9.6.4
Cancel the common factor.
Step 9.6.5
Rewrite the expression.
Step 9.7
Since is constant with respect to , the derivative of with respect to is .
Step 10
Differentiate using the Product Rule which states that is where and .
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
Differentiate using the Power Rule which states that is where .
Step 11.3
Replace all occurrences of with .
Step 12
To write as a fraction with a common denominator, multiply by .
Step 13
Combine and .
Step 14
Combine the numerators over the common denominator.
Step 15
Simplify the numerator.
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Step 15.1
Multiply by .
Step 15.2
Subtract from .
Step 16
Combine fractions.
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Step 16.1
Move the negative in front of the fraction.
Step 16.2
Combine and .
Step 16.3
Move to the denominator using the negative exponent rule .
Step 16.4
Combine and .
Step 17
By the Sum Rule, the derivative of with respect to is .
Step 18
Differentiate using the Power Rule which states that is where .
Step 19
Since is constant with respect to , the derivative of with respect to is .
Step 20
Combine fractions.
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Step 20.1
Add and .
Step 20.2
Multiply by .
Step 20.3
Combine and .
Step 20.4
Combine and .
Step 21
Raise to the power of .
Step 22
Use the power rule to combine exponents.
Step 23
Add and .
Step 24
Factor out of .
Step 25
Cancel the common factors.
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Step 25.1
Factor out of .
Step 25.2
Cancel the common factor.
Step 25.3
Rewrite the expression.
Step 26
Move the negative in front of the fraction.
Step 27
Differentiate using the Power Rule which states that is where .
Step 28
Move to the left of .
Step 29
Combine and using a common denominator.
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Step 29.1
Move .
Step 29.2
To write as a fraction with a common denominator, multiply by .
Step 29.3
Combine the numerators over the common denominator.
Step 30
Multiply by by adding the exponents.
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Step 30.1
Move .
Step 30.2
Use the power rule to combine exponents.
Step 30.3
Combine the numerators over the common denominator.
Step 30.4
Subtract from .
Step 30.5
Divide by .
Step 31
Simplify .
Step 32
By the Sum Rule, the derivative of with respect to is .
Step 33
Differentiate using the Power Rule which states that is where .
Step 34
Since is constant with respect to , the derivative of with respect to is .
Step 35
Simplify the expression.
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Step 35.1
Add and .
Step 35.2
Multiply by .
Step 36
Simplify.
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Step 36.1
Rewrite the expression using the negative exponent rule .
Step 36.2
Apply the distributive property.
Step 36.3
Apply the distributive property.
Step 36.4
Apply the distributive property.
Step 36.5
Simplify the numerator.
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Step 36.5.1
Simplify the numerator.
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Step 36.5.1.1
Factor out of .
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Step 36.5.1.1.1
Factor out of .
Step 36.5.1.1.2
Factor out of .
Step 36.5.1.1.3
Factor out of .
Step 36.5.1.1.4
Factor out of .
Step 36.5.1.1.5
Factor out of .
Step 36.5.1.2
Multiply by .
Step 36.5.1.3
Add and .
Step 36.5.2
To write as a fraction with a common denominator, multiply by .
Step 36.5.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 36.5.3.1
Multiply by .
Step 36.5.3.2
Multiply by by adding the exponents.
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Step 36.5.3.2.1
Use the power rule to combine exponents.
Step 36.5.3.2.2
Combine the numerators over the common denominator.
Step 36.5.3.2.3
Add and .
Step 36.5.4
Combine the numerators over the common denominator.
Step 36.5.5
Simplify the numerator.
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Step 36.5.5.1
Factor out of .
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Step 36.5.5.1.1
Factor out of .
Step 36.5.5.1.2
Factor out of .
Step 36.5.5.1.3
Factor out of .
Step 36.5.5.2
Divide by .
Step 36.5.5.3
Simplify.
Step 36.5.5.4
Apply the distributive property.
Step 36.5.5.5
Multiply by .
Step 36.5.5.6
Subtract from .
Step 36.5.5.7
Add and .
Step 36.5.5.8
Subtract from .
Step 36.5.6
Move to the left of .
Step 36.5.7
Move the negative in front of the fraction.
Step 36.5.8
Apply the distributive property.
Step 36.5.9
Rewrite using the commutative property of multiplication.
Step 36.5.10
Multiply .
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Step 36.5.10.1
Multiply by .
Step 36.5.10.2
Combine and .
Step 36.5.10.3
Multiply by .
Step 36.5.11
Simplify each term.
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Step 36.5.11.1
Multiply .
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Step 36.5.11.1.1
Combine and .
Step 36.5.11.1.2
Multiply by by adding the exponents.
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Step 36.5.11.1.2.1
Move .
Step 36.5.11.1.2.2
Multiply by .
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Step 36.5.11.1.2.2.1
Raise to the power of .
Step 36.5.11.1.2.2.2
Use the power rule to combine exponents.
Step 36.5.11.1.2.3
Add and .
Step 36.5.11.2
Move the negative in front of the fraction.
Step 36.5.12
Multiply by by adding the exponents.
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Step 36.5.12.1
Move .
Step 36.5.12.2
Multiply by .
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Step 36.5.12.2.1
Raise to the power of .
Step 36.5.12.2.2
Use the power rule to combine exponents.
Step 36.5.12.3
Add and .
Step 36.5.13
Combine and .
Step 36.5.14
Multiply .
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Step 36.5.14.1
Multiply by .
Step 36.5.14.2
Combine and .
Step 36.5.15
To write as a fraction with a common denominator, multiply by .
Step 36.5.16
Combine and .
Step 36.5.17
Combine the numerators over the common denominator.
Step 36.5.18
Reorder terms.
Step 36.5.19
Simplify the numerator.
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Step 36.5.19.1
Factor out of .
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Step 36.5.19.1.1
Factor out of .
Step 36.5.19.1.2
Factor out of .
Step 36.5.19.1.3
Factor out of .
Step 36.5.19.2
Multiply by by adding the exponents.
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Step 36.5.19.2.1
Move .
Step 36.5.19.2.2
Use the power rule to combine exponents.
Step 36.5.19.2.3
Combine the numerators over the common denominator.
Step 36.5.19.2.4
Add and .
Step 36.5.19.2.5
Divide by .
Step 36.5.19.3
Simplify each term.
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Step 36.5.19.3.1
Rewrite as .
Step 36.5.19.3.2
Expand using the FOIL Method.
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Step 36.5.19.3.2.1
Apply the distributive property.
Step 36.5.19.3.2.2
Apply the distributive property.
Step 36.5.19.3.2.3
Apply the distributive property.
Step 36.5.19.3.3
Simplify and combine like terms.
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Step 36.5.19.3.3.1
Simplify each term.
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Step 36.5.19.3.3.1.1
Multiply by by adding the exponents.
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Step 36.5.19.3.3.1.1.1
Use the power rule to combine exponents.
Step 36.5.19.3.3.1.1.2
Add and .
Step 36.5.19.3.3.1.2
Move to the left of .
Step 36.5.19.3.3.1.3
Multiply by .
Step 36.5.19.3.3.2
Add and .
Step 36.5.19.3.4
Apply the distributive property.
Step 36.5.19.3.5
Simplify.
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Step 36.5.19.3.5.1
Multiply by .
Step 36.5.19.3.5.2
Multiply by .
Step 36.5.19.4
Subtract from .
Step 36.5.19.5
Rewrite as .
Step 36.5.19.6
Let . Substitute for all occurrences of .
Step 36.5.19.7
Factor by grouping.
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Step 36.5.19.7.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 36.5.19.7.1.1
Factor out of .
Step 36.5.19.7.1.2
Rewrite as plus
Step 36.5.19.7.1.3
Apply the distributive property.
Step 36.5.19.7.2
Factor out the greatest common factor from each group.
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Step 36.5.19.7.2.1
Group the first two terms and the last two terms.
Step 36.5.19.7.2.2
Factor out the greatest common factor (GCF) from each group.
Step 36.5.19.7.3
Factor the polynomial by factoring out the greatest common factor, .
Step 36.5.19.8
Replace all occurrences of with .
Step 36.5.20
Combine the numerators over the common denominator.
Step 36.5.21
Simplify each term.
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Step 36.5.21.1
Apply the distributive property.
Step 36.5.21.2
Rewrite using the commutative property of multiplication.
Step 36.5.21.3
Multiply by .
Step 36.5.21.4
Simplify each term.
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Step 36.5.21.4.1
Multiply by by adding the exponents.
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Step 36.5.21.4.1.1
Move .
Step 36.5.21.4.1.2
Multiply by .
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Step 36.5.21.4.1.2.1
Raise to the power of .
Step 36.5.21.4.1.2.2
Use the power rule to combine exponents.
Step 36.5.21.4.1.3
Add and .
Step 36.5.21.4.2
Multiply by .
Step 36.5.21.5
Expand using the FOIL Method.
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Step 36.5.21.5.1
Apply the distributive property.
Step 36.5.21.5.2
Apply the distributive property.
Step 36.5.21.5.3
Apply the distributive property.
Step 36.5.21.6
Simplify and combine like terms.
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Step 36.5.21.6.1
Simplify each term.
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Step 36.5.21.6.1.1
Multiply by by adding the exponents.
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Step 36.5.21.6.1.1.1
Move .
Step 36.5.21.6.1.1.2
Use the power rule to combine exponents.
Step 36.5.21.6.1.1.3
Add and .
Step 36.5.21.6.1.2
Multiply by .
Step 36.5.21.6.1.3
Multiply by by adding the exponents.
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Step 36.5.21.6.1.3.1
Move .
Step 36.5.21.6.1.3.2
Multiply by .
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Step 36.5.21.6.1.3.2.1
Raise to the power of .
Step 36.5.21.6.1.3.2.2
Use the power rule to combine exponents.
Step 36.5.21.6.1.3.3
Add and .
Step 36.5.21.6.1.4
Multiply by .
Step 36.5.21.6.2
Subtract from .
Step 36.5.22
Subtract from .
Step 36.5.23
Simplify the numerator.
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Step 36.5.23.1
Factor out of .
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Step 36.5.23.1.1
Factor out of .
Step 36.5.23.1.2
Factor out of .
Step 36.5.23.1.3
Factor out of .
Step 36.5.23.1.4
Factor out of .
Step 36.5.23.1.5
Factor out of .
Step 36.5.23.2
Rewrite as .
Step 36.5.23.3
Let . Substitute for all occurrences of .
Step 36.5.23.4
Factor by grouping.
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Step 36.5.23.4.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 36.5.23.4.1.1
Factor out of .
Step 36.5.23.4.1.2
Rewrite as plus
Step 36.5.23.4.1.3
Apply the distributive property.
Step 36.5.23.4.2
Factor out the greatest common factor from each group.
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Step 36.5.23.4.2.1
Group the first two terms and the last two terms.
Step 36.5.23.4.2.2
Factor out the greatest common factor (GCF) from each group.
Step 36.5.23.4.3
Factor the polynomial by factoring out the greatest common factor, .
Step 36.5.23.5
Replace all occurrences of with .
Step 36.5.24
To write as a fraction with a common denominator, multiply by .
Step 36.5.25
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 36.5.25.1
Multiply by .
Step 36.5.25.2
Multiply by by adding the exponents.
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Step 36.5.25.2.1
Use the power rule to combine exponents.
Step 36.5.25.2.2
Combine the numerators over the common denominator.
Step 36.5.25.2.3
Add and .
Step 36.5.26
Combine the numerators over the common denominator.
Step 36.5.27
Cancel the common factor of .
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Step 36.5.27.1
Cancel the common factor.
Step 36.5.27.2
Rewrite the expression.
Step 36.5.28
Simplify the numerator.
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Step 36.5.28.1
Factor out of .
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Step 36.5.28.1.1
Factor out of .
Step 36.5.28.1.2
Factor out of .
Step 36.5.28.1.3
Factor out of .
Step 36.5.28.2
Simplify.
Step 36.5.28.3
Apply the distributive property.
Step 36.5.28.4
Multiply by by adding the exponents.
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Step 36.5.28.4.1
Use the power rule to combine exponents.
Step 36.5.28.4.2
Add and .
Step 36.5.28.5
Move to the left of .
Step 36.5.28.6
Expand using the FOIL Method.
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Step 36.5.28.6.1
Apply the distributive property.
Step 36.5.28.6.2
Apply the distributive property.
Step 36.5.28.6.3
Apply the distributive property.
Step 36.5.28.7
Simplify and combine like terms.
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Step 36.5.28.7.1
Simplify each term.
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Step 36.5.28.7.1.1
Multiply by by adding the exponents.
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Step 36.5.28.7.1.1.1
Move .
Step 36.5.28.7.1.1.2
Use the power rule to combine exponents.
Step 36.5.28.7.1.1.3
Add and .
Step 36.5.28.7.1.2
Multiply by .
Step 36.5.28.7.1.3
Multiply by .
Step 36.5.28.7.2
Subtract from .
Step 36.5.28.8
Subtract from .
Step 36.5.28.9
Add and .
Step 36.5.28.10
Subtract from .
Step 36.5.28.11
Factor out of .
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Step 36.5.28.11.1
Factor out of .
Step 36.5.28.11.2
Factor out of .
Step 36.5.28.11.3
Factor out of .
Step 36.5.28.12
Multiply by .
Step 36.6
Combine terms.
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Step 36.6.1
Rewrite as a product.
Step 36.6.2
Multiply by .
Step 36.6.3
Multiply by by adding the exponents.
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Step 36.6.3.1
Use the power rule to combine exponents.
Step 36.6.3.2
To write as a fraction with a common denominator, multiply by .
Step 36.6.3.3
Combine and .
Step 36.6.3.4
Combine the numerators over the common denominator.
Step 36.6.3.5
Simplify the numerator.
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Step 36.6.3.5.1
Multiply by .
Step 36.6.3.5.2
Add and .
Step 36.7
Factor out of .
Step 36.8
Rewrite as .
Step 36.9
Factor out of .
Step 36.10
Rewrite as .
Step 36.11
Move the negative in front of the fraction.