Calculus Examples

Find the Derivative - d/dx ( square root of x+x)/(x^2)
Step 1
Use to rewrite as .
Step 2
Factor out of .
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Step 2.1
Multiply by .
Step 2.2
Raise to the power of .
Step 2.3
Factor out of .
Step 2.4
Factor out of .
Step 3
Move to the denominator using the negative exponent rule .
Step 4
Multiply by by adding the exponents.
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Step 4.1
Use the power rule to combine exponents.
Step 4.2
To write as a fraction with a common denominator, multiply by .
Step 4.3
Combine and .
Step 4.4
Combine the numerators over the common denominator.
Step 4.5
Simplify the numerator.
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Step 4.5.1
Multiply by .
Step 4.5.2
Subtract from .
Step 5
Differentiate using the Quotient Rule which states that is where and .
Step 6
Differentiate.
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Step 6.1
Multiply the exponents in .
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Step 6.1.1
Apply the power rule and multiply exponents, .
Step 6.1.2
Cancel the common factor of .
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Step 6.1.2.1
Cancel the common factor.
Step 6.1.2.2
Rewrite the expression.
Step 6.2
By the Sum Rule, the derivative of with respect to is .
Step 6.3
Since is constant with respect to , the derivative of with respect to is .
Step 6.4
Add and .
Step 6.5
Differentiate using the Power Rule which states that is where .
Step 7
To write as a fraction with a common denominator, multiply by .
Step 8
Combine and .
Step 9
Combine the numerators over the common denominator.
Step 10
Simplify the numerator.
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Step 10.1
Multiply by .
Step 10.2
Subtract from .
Step 11
Move the negative in front of the fraction.
Step 12
Combine and .
Step 13
Combine and .
Step 14
Multiply by by adding the exponents.
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Step 14.1
Use the power rule to combine exponents.
Step 14.2
Combine the numerators over the common denominator.
Step 14.3
Subtract from .
Step 14.4
Divide by .
Step 15
Simplify .
Step 16
Multiply by .
Step 17
Simplify terms.
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Step 17.1
Combine.
Step 17.2
Apply the distributive property.
Step 17.3
Cancel the common factor of .
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Step 17.3.1
Cancel the common factor.
Step 17.3.2
Rewrite the expression.
Step 17.4
Multiply by .
Step 18
Differentiate using the Power Rule which states that is where .
Step 19
To write as a fraction with a common denominator, multiply by .
Step 20
Combine and .
Step 21
Combine the numerators over the common denominator.
Step 22
Simplify the numerator.
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Step 22.1
Multiply by .
Step 22.2
Subtract from .
Step 23
Combine and .
Step 24
Combine and .
Step 25
Multiply by .
Step 26
Factor out of .
Step 27
Cancel the common factors.
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Step 27.1
Factor out of .
Step 27.2
Cancel the common factor.
Step 27.3
Rewrite the expression.
Step 27.4
Divide by .
Step 28
Simplify.
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Step 28.1
Apply the distributive property.
Step 28.2
Simplify the numerator.
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Step 28.2.1
Simplify each term.
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Step 28.2.1.1
Multiply by .
Step 28.2.1.2
Multiply by by adding the exponents.
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Step 28.2.1.2.1
Move .
Step 28.2.1.2.2
Use the power rule to combine exponents.
Step 28.2.1.2.3
Combine the numerators over the common denominator.
Step 28.2.1.2.4
Add and .
Step 28.2.1.2.5
Divide by .
Step 28.2.1.3
Simplify .
Step 28.2.2
Subtract from .
Step 28.3
Simplify the numerator.
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Step 28.3.1
Factor out of .
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Step 28.3.1.1
Factor out of .
Step 28.3.1.2
Factor out of .
Step 28.3.1.3
Factor out of .
Step 28.3.2
Factor out of .
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Step 28.3.2.1
Factor out of .
Step 28.3.2.2
Rewrite as .
Step 28.3.2.3
Factor out of .
Step 28.3.3
Factor out negative.
Step 28.4
Move to the denominator using the negative exponent rule .
Step 28.5
Multiply by by adding the exponents.
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Step 28.5.1
Move .
Step 28.5.2
Use the power rule to combine exponents.
Step 28.5.3
To write as a fraction with a common denominator, multiply by .
Step 28.5.4
Combine and .
Step 28.5.5
Combine the numerators over the common denominator.
Step 28.5.6
Simplify the numerator.
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Step 28.5.6.1
Multiply by .
Step 28.5.6.2
Add and .
Step 28.6
Move the negative in front of the fraction.