Calculus Examples

Find the Derivative - d/dx f(x)=(2x)/(x+ square root of x)
Step 1
Differentiate using the Constant Multiple Rule.
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Step 1.1
Use to rewrite as .
Step 1.2
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
Differentiate.
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Step 3.1
Differentiate using the Power Rule which states that is where .
Step 3.2
Multiply by .
Step 3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
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
Move the negative in front of the fraction.
Step 9
Combine and .
Step 10
Move to the denominator using the negative exponent rule .
Step 11
Combine and .
Step 12
Simplify.
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Step 12.1
Apply the distributive property.
Step 12.2
Apply the distributive property.
Step 12.3
Simplify the numerator.
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Step 12.3.1
Simplify each term.
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Step 12.3.1.1
Multiply by .
Step 12.3.1.2
Multiply by .
Step 12.3.1.3
Combine and .
Step 12.3.1.4
Cancel the common factor of .
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Step 12.3.1.4.1
Move the leading negative in into the numerator.
Step 12.3.1.4.2
Cancel the common factor.
Step 12.3.1.4.3
Rewrite the expression.
Step 12.3.1.5
Move to the numerator using the negative exponent rule .
Step 12.3.1.6
Multiply by by adding the exponents.
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Step 12.3.1.6.1
Move .
Step 12.3.1.6.2
Multiply by .
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Step 12.3.1.6.2.1
Raise to the power of .
Step 12.3.1.6.2.2
Use the power rule to combine exponents.
Step 12.3.1.6.3
Write as a fraction with a common denominator.
Step 12.3.1.6.4
Combine the numerators over the common denominator.
Step 12.3.1.6.5
Add and .
Step 12.3.2
Combine the opposite terms in .
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Step 12.3.2.1
Subtract from .
Step 12.3.2.2
Add and .
Step 12.3.3
Subtract from .
Step 12.4
Simplify the denominator.
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Step 12.4.1
Factor out of .
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Step 12.4.1.1
Raise to the power of .
Step 12.4.1.2
Factor out of .
Step 12.4.1.3
Multiply by .
Step 12.4.1.4
Factor out of .
Step 12.4.2
Apply the product rule to .
Step 12.4.3
Multiply the exponents in .
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Step 12.4.3.1
Apply the power rule and multiply exponents, .
Step 12.4.3.2
Cancel the common factor of .
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Step 12.4.3.2.1
Cancel the common factor.
Step 12.4.3.2.2
Rewrite the expression.
Step 12.4.4
Simplify.
Step 12.5
Move to the denominator using the negative exponent rule .
Step 12.6
Multiply by by adding the exponents.
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Step 12.6.1
Move .
Step 12.6.2
Multiply by .
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Step 12.6.2.1
Raise to the power of .
Step 12.6.2.2
Use the power rule to combine exponents.
Step 12.6.3
Write as a fraction with a common denominator.
Step 12.6.4
Combine the numerators over the common denominator.
Step 12.6.5
Add and .