# Find dy/dx (xy)^(1/2)=x-2y

Differentiate both sides of the equation.
Differentiate the left side of the equation.
Differentiate using the chain rule, which states that is where and .
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.
Multiply by .
Subtract from .
Combine fractions.
Move the negative in front of the fraction.
Combine and .
Move to the denominator using the negative exponent rule .
Differentiate using the Product Rule which states that is where and .
Rewrite as .
Differentiate using the Power Rule which states that is where .
Multiply by .
Simplify.
Apply the product rule to .
Apply the distributive property.
Combine terms.
Combine and .
Move to the numerator using the negative exponent rule .
Multiply by by adding the exponents.
Multiply by .
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.
Subtract from .
Combine and .
Move to the numerator using the negative exponent rule .
Multiply by by adding the exponents.
Multiply by .
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.
Subtract from .
Reorder terms.
Differentiate the right side of the equation.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Rewrite as .
Reform the equation by setting the left side equal to the right side.
Solve for .
Combine and .
Add to both sides of the equation.
Find the LCD of the terms in the equation.
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.
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.
Multiply each term in by in order to remove all the denominators from the equation.
Simplify each term.
Rewrite using the commutative property of multiplication.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Add and .
Divide by .
Simplify .
Rewrite using the commutative property of multiplication.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by by adding the exponents.
Use the power rule to combine exponents.
Combine the numerators over the common denominator.
Add and .
Divide by .
Simplify .
Multiply by .
Multiply by .
Solve the equation.
Subtract from both sides of the equation.
Factor out of .
Move .
Factor out of .
Factor out of .
Factor out of .
Divide each term by and simplify.
Divide each term in by .
Simplify .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor.
Divide by .
Simplify .
Simplify each term.
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Reorder the factors of .
Combine the numerators over the common denominator.
Factor out of .
Move .
Factor out of .
Factor out of .
Factor out of .
Replace with .
Find dy/dx (xy)^(1/2)=x-2y

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