Find dx/dy y=(x^2+8x+3)/( square root of x)

Math
Use to rewrite as .
Differentiate both sides of the equation.
Differentiate using the Power Rule which states that is where .
Differentiate the right side of the equation.
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Differentiate using the Quotient Rule which states that is where and .
Multiply the exponents in .
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Apply the power rule and multiply exponents, .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Simplify.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Rewrite as .
Since is constant with respect to , the derivative of with respect to is .
Rewrite as .
Differentiate using the Constant Rule.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Move to the denominator using the negative exponent rule .
Rewrite as .
Simplify.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify each term.
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Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Add and .
Rewrite using the commutative property of multiplication.
Cancel the common factor of .
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Combine and .
Multiply by .
Cancel the common factor of .
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Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Combine and .
Move to the numerator using the negative exponent rule .
Multiply by by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Add and .
Move to the left of .
Multiply by .
Combine and .
Move the negative in front of the fraction.
Combine and .
Combine terms.
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Rewrite as .
Rewrite as .
Rewrite as .
Reorder terms.
Reform the equation by setting the left side equal to the right side.
Solve for .
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Rewrite the equation as .
Multiply both sides of the equation by .
Simplify .
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Remove parentheses.
To write as a fraction with a common denominator, multiply by .
Simplify terms.
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Combine and .
Combine the numerators over the common denominator.
Simplify each term.
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Simplify the numerator.
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Factor out of .
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Reorder the expression.
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Move .
Move .
Move .
Factor out of .
Factor out of .
Factor out of .
Multiply by .
Subtract from .
Move to the left of .
Simplify by adding terms.
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Subtract from .
Reorder factors in .
Multiply by .
Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Since has no factors besides and .
is a prime number
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
The LCM for is the numeric part multiplied by the variable part.
Multiply each term by and simplify.
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Multiply each term in by in order to remove all the denominators from the equation.
Simplify each term.
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Multiply by by adding the exponents.
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Move .
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Add and .
Divide by .
Simplify .
Multiply by .
Rewrite using the commutative property of multiplication.
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Multiply by by adding the exponents.
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Move .
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Add and .
Divide by .
Cancel the common factor of .
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Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Simplify .
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Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Add and .
Solve the equation.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Let . Substitute for all occurrences of .
Factor by grouping.
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Reorder terms.
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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Factor out of .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
Factor.
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Replace all occurrences of with .
Remove unnecessary parentheses.
Divide each term by and simplify.
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Divide each term in by .
Simplify .
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Replace with .
Find dx/dy y=(x^2+8x+3)/( square root of x)


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