Calculus Examples

Find dy/dx y=( square root of x-4)/( square root of x+4)
Step 1
Rewrite the right side 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
The derivative of with respect to is .
Step 4
Differentiate the right side of the equation.
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Step 4.1
Differentiate using the Quotient Rule which states that is where and .
Step 4.2
Differentiate.
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Step 4.2.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2.2
Differentiate using the Power Rule which states that is where .
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
Combine fractions.
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Step 4.7.1
Move the negative in front of the fraction.
Step 4.7.2
Combine and .
Step 4.7.3
Move to the denominator using the negative exponent rule .
Step 4.8
Since is constant with respect to , the derivative of with respect to is .
Step 4.9
Add and .
Step 4.10
By the Sum Rule, the derivative of with respect to is .
Step 4.11
Differentiate using the Power Rule which states that is where .
Step 4.12
To write as a fraction with a common denominator, multiply by .
Step 4.13
Combine and .
Step 4.14
Combine the numerators over the common denominator.
Step 4.15
Simplify the numerator.
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Step 4.15.1
Multiply by .
Step 4.15.2
Subtract from .
Step 4.16
Combine fractions.
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Step 4.16.1
Move the negative in front of the fraction.
Step 4.16.2
Combine and .
Step 4.16.3
Move to the denominator using the negative exponent rule .
Step 4.17
Since is constant with respect to , the derivative of with respect to is .
Step 4.18
Add and .
Step 4.19
Simplify.
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Step 4.19.1
Apply the distributive property.
Step 4.19.2
Apply the distributive property.
Step 4.19.3
Apply the distributive property.
Step 4.19.4
Simplify the numerator.
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Step 4.19.4.1
Combine the opposite terms in .
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Step 4.19.4.1.1
Subtract from .
Step 4.19.4.1.2
Add and .
Step 4.19.4.2
Simplify each term.
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Step 4.19.4.2.1
Cancel the common factor of .
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Step 4.19.4.2.1.1
Factor out of .
Step 4.19.4.2.1.2
Cancel the common factor.
Step 4.19.4.2.1.3
Rewrite the expression.
Step 4.19.4.2.2
Combine and .
Step 4.19.4.2.3
Multiply by .
Step 4.19.4.2.4
Cancel the common factor of .
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Step 4.19.4.2.4.1
Factor out of .
Step 4.19.4.2.4.2
Cancel the common factor.
Step 4.19.4.2.4.3
Rewrite the expression.
Step 4.19.4.2.5
Combine and .
Step 4.19.4.3
Combine the numerators over the common denominator.
Step 4.19.4.4
Add and .
Step 4.19.5
Combine terms.
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Step 4.19.5.1
Rewrite as a product.
Step 4.19.5.2
Multiply by .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .