Calculus Examples

Find the Derivative - d/dx ( square root of x)/(x^3+1)
Step 1
Use to rewrite as .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Differentiate using the Power Rule which states that is where .
Step 4
To write as a fraction with a common denominator, multiply by .
Step 5
Combine and .
Step 6
Combine the numerators over the common denominator.
Step 7
Simplify the numerator.
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Step 7.1
Multiply by .
Step 7.2
Subtract from .
Step 8
Combine fractions.
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Step 8.1
Move the negative in front of the fraction.
Step 8.2
Combine and .
Step 8.3
Move to the denominator using the negative exponent rule .
Step 9
By the Sum Rule, the derivative of with respect to is .
Step 10
Differentiate using the Power Rule which states that is where .
Step 11
Since is constant with respect to , the derivative of with respect to is .
Step 12
Simplify the expression.
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Step 12.1
Add and .
Step 12.2
Multiply by .
Step 13
Multiply by by adding the exponents.
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Step 13.1
Move .
Step 13.2
Use the power rule to combine exponents.
Step 13.3
To write as a fraction with a common denominator, multiply by .
Step 13.4
Combine and .
Step 13.5
Combine the numerators over the common denominator.
Step 13.6
Simplify the numerator.
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Step 13.6.1
Multiply by .
Step 13.6.2
Add and .
Step 14
Simplify.
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Step 14.1
Apply the distributive property.
Step 14.2
Simplify the numerator.
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Step 14.2.1
Simplify each term.
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Step 14.2.1.1
Cancel the common factor of .
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Step 14.2.1.1.1
Factor out of .
Step 14.2.1.1.2
Factor out of .
Step 14.2.1.1.3
Cancel the common factor.
Step 14.2.1.1.4
Rewrite the expression.
Step 14.2.1.2
Combine and .
Step 14.2.1.3
Multiply by .
Step 14.2.2
To write as a fraction with a common denominator, multiply by .
Step 14.2.3
Combine and .
Step 14.2.4
Combine the numerators over the common denominator.
Step 14.2.5
Simplify each term.
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Step 14.2.5.1
Simplify the numerator.
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Step 14.2.5.1.1
Factor out of .
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Step 14.2.5.1.1.1
Move .
Step 14.2.5.1.1.2
Multiply by .
Step 14.2.5.1.1.3
Factor out of .
Step 14.2.5.1.1.4
Factor out of .
Step 14.2.5.1.2
Multiply by .
Step 14.2.5.1.3
Subtract from .
Step 14.2.5.2
Move to the left of .
Step 14.2.5.3
Move the negative in front of the fraction.
Step 14.3
Combine terms.
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Step 14.3.1
Multiply by .
Step 14.3.2
Combine.
Step 14.3.3
Apply the distributive property.
Step 14.3.4
Cancel the common factor of .
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Step 14.3.4.1
Cancel the common factor.
Step 14.3.4.2
Rewrite the expression.
Step 14.3.5
Multiply by .
Step 14.3.6
Combine and .
Step 14.3.7
Multiply by .
Step 14.3.8
Combine and .
Step 14.3.9
Multiply by by adding the exponents.
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Step 14.3.9.1
Move .
Step 14.3.9.2
Use the power rule to combine exponents.
Step 14.3.9.3
Combine the numerators over the common denominator.
Step 14.3.9.4
Add and .
Step 14.3.9.5
Divide by .
Step 14.3.10
Move to the left of .
Step 14.3.11
Cancel the common factor of and .
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Step 14.3.11.1
Factor out of .
Step 14.3.11.2
Cancel the common factors.
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Step 14.3.11.2.1
Factor out of .
Step 14.3.11.2.2
Cancel the common factor.
Step 14.3.11.2.3
Rewrite the expression.
Step 14.3.11.2.4
Divide by .
Step 14.4
Simplify the denominator.
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Step 14.4.1
Rewrite as .
Step 14.4.2
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 14.4.3
Simplify.
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Step 14.4.3.1
Multiply by .
Step 14.4.3.2
One to any power is one.
Step 14.4.4
Apply the product rule to .
Step 14.5
Factor out of .
Step 14.6
Rewrite as .
Step 14.7
Factor out of .
Step 14.8
Rewrite as .
Step 14.9
Move the negative in front of the fraction.