Calculus Examples

Find the Derivative - d/dx (1+ square root of 3x)/(1- square root of 3x)
Step 1
Simplify with factoring out.
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Step 1.1
Use to rewrite as .
Step 1.2
Factor out of .
Step 1.3
Apply basic rules of exponents.
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Step 1.3.1
Apply the product rule to .
Step 1.3.2
Use to rewrite as .
Step 1.4
Factor out of .
Step 1.5
Apply the product rule to .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Differentiate.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Add and .
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
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
Combine fractions.
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Step 8.1
Move the negative in front of the fraction.
Step 8.2
Combine and .
Step 8.3
Combine and .
Step 8.4
Move to the denominator using the negative exponent rule .
Step 9
By the Sum Rule, the derivative of with respect to is .
Step 10
Since is constant with respect to , the derivative of with respect to is .
Step 11
Add and .
Step 12
Since is constant with respect to , the derivative of with respect to is .
Step 13
Multiply.
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Step 13.1
Multiply by .
Step 13.2
Multiply by .
Step 14
Differentiate using the Power Rule which states that is where .
Step 15
To write as a fraction with a common denominator, multiply by .
Step 16
Combine and .
Step 17
Combine the numerators over the common denominator.
Step 18
Simplify the numerator.
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Step 18.1
Multiply by .
Step 18.2
Subtract from .
Step 19
Move the negative in front of the fraction.
Step 20
Combine and .
Step 21
Combine and .
Step 22
Move to the denominator using the negative exponent rule .
Step 23
Simplify.
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Step 23.1
Apply the distributive property.
Step 23.2
Apply the distributive property.
Step 23.3
Simplify the numerator.
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Step 23.3.1
Combine the opposite terms in .
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Step 23.3.1.1
Add and .
Step 23.3.1.2
Add and .
Step 23.3.2
Simplify each term.
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Step 23.3.2.1
Multiply by .
Step 23.3.2.2
Multiply by .
Step 23.3.3
Combine the numerators over the common denominator.
Step 23.3.4
Add and .
Step 23.3.5
Cancel the common factor.
Step 23.3.6
Rewrite the expression.
Step 23.4
Combine terms.
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Step 23.4.1
Rewrite as a product.
Step 23.4.2
Multiply by .