Calculus Examples

Find dy/dx square root of x=3 square root of y
Step 1
Rewrite the equation with rational exponents.
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Step 1.1
Use to rewrite as .
Step 1.2
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
Differentiate the left side of the equation.
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Step 3.1
Differentiate using the Power Rule which states that is where .
Step 3.2
To write as a fraction with a common denominator, multiply by .
Step 3.3
Combine and .
Step 3.4
Combine the numerators over the common denominator.
Step 3.5
Simplify the numerator.
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Step 3.5.1
Multiply by .
Step 3.5.2
Subtract from .
Step 3.6
Move the negative in front of the fraction.
Step 3.7
Simplify.
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Step 3.7.1
Rewrite the expression using the negative exponent rule .
Step 3.7.2
Multiply by .
Step 4
Differentiate the right side of the equation.
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Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the chain rule, which states that is where and .
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Step 4.2.1
To apply the Chain Rule, set as .
Step 4.2.2
Differentiate using the Power Rule which states that is where .
Step 4.2.3
Replace all occurrences of with .
Step 4.3
To write as a fraction with a common denominator, multiply by .
Step 4.4
Combine and .
Step 4.5
Combine the numerators over the common denominator.
Step 4.6
Simplify the numerator.
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Step 4.6.1
Multiply by .
Step 4.6.2
Subtract from .
Step 4.7
Move the negative in front of the fraction.
Step 4.8
Combine and .
Step 4.9
Move to the denominator using the negative exponent rule .
Step 4.10
Combine and .
Step 4.11
Rewrite as .
Step 4.12
Combine and .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Solve for .
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Step 6.1
Rewrite the equation as .
Step 6.2
Multiply both sides by .
Step 6.3
Simplify.
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Step 6.3.1
Simplify the left side.
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Step 6.3.1.1
Simplify .
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Step 6.3.1.1.1
Rewrite using the commutative property of multiplication.
Step 6.3.1.1.2
Cancel the common factor of .
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Step 6.3.1.1.2.1
Cancel the common factor.
Step 6.3.1.1.2.2
Rewrite the expression.
Step 6.3.1.1.3
Cancel the common factor of .
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Step 6.3.1.1.3.1
Cancel the common factor.
Step 6.3.1.1.3.2
Rewrite the expression.
Step 6.3.2
Simplify the right side.
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Step 6.3.2.1
Simplify .
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Step 6.3.2.1.1
Rewrite using the commutative property of multiplication.
Step 6.3.2.1.2
Cancel the common factor of .
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Step 6.3.2.1.2.1
Cancel the common factor.
Step 6.3.2.1.2.2
Rewrite the expression.
Step 6.3.2.1.3
Combine and .
Step 6.4
Divide each term in by and simplify.
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Step 6.4.1
Divide each term in by .
Step 6.4.2
Simplify the left side.
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Step 6.4.2.1
Cancel the common factor of .
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Step 6.4.2.1.1
Cancel the common factor.
Step 6.4.2.1.2
Divide by .
Step 6.4.3
Simplify the right side.
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Step 6.4.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.4.3.2
Combine.
Step 6.4.3.3
Simplify the expression.
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Step 6.4.3.3.1
Multiply by .
Step 6.4.3.3.2
Move to the left of .
Step 7
Replace with .