Calculus Examples

Find dy/dx y=(x^2+8x+3)/( square root of x)
Step 1
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
Multiply the exponents in .
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Step 4.2.1
Apply the power rule and multiply exponents, .
Step 4.2.2
Cancel the common factor of .
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Step 4.2.2.1
Cancel the common factor.
Step 4.2.2.2
Rewrite the expression.
Step 4.3
Simplify.
Step 4.4
By the Sum Rule, the derivative of with respect to is .
Step 4.5
Differentiate using the Power Rule which states that is where .
Step 4.6
Since is constant with respect to , the derivative of with respect to is .
Step 4.7
Differentiate using the Power Rule which states that is where .
Step 4.8
Multiply by .
Step 4.9
Since is constant with respect to , the derivative of with respect to is .
Step 4.10
Add and .
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
Move the negative in front of the fraction.
Step 4.17
Combine and .
Step 4.18
Move to the denominator using the negative exponent rule .
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
Simplify each term.
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Step 4.19.4.1.1
Rewrite using the commutative property of multiplication.
Step 4.19.4.1.2
Multiply by by adding the exponents.
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Step 4.19.4.1.2.1
Move .
Step 4.19.4.1.2.2
Multiply by .
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Step 4.19.4.1.2.2.1
Raise to the power of .
Step 4.19.4.1.2.2.2
Use the power rule to combine exponents.
Step 4.19.4.1.2.3
Write as a fraction with a common denominator.
Step 4.19.4.1.2.4
Combine the numerators over the common denominator.
Step 4.19.4.1.2.5
Add and .
Step 4.19.4.1.3
Move to the left of .
Step 4.19.4.1.4
Cancel the common factor of .
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Step 4.19.4.1.4.1
Factor out of .
Step 4.19.4.1.4.2
Factor out of .
Step 4.19.4.1.4.3
Cancel the common factor.
Step 4.19.4.1.4.4
Rewrite the expression.
Step 4.19.4.1.5
Combine and .
Step 4.19.4.1.6
Multiply by .
Step 4.19.4.1.7
Cancel the common factor of .
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Step 4.19.4.1.7.1
Factor out of .
Step 4.19.4.1.7.2
Factor out of .
Step 4.19.4.1.7.3
Cancel the common factor.
Step 4.19.4.1.7.4
Rewrite the expression.
Step 4.19.4.1.8
Combine and .
Step 4.19.4.1.9
Combine and .
Step 4.19.4.1.10
Move to the numerator using the negative exponent rule .
Step 4.19.4.1.11
Multiply by by adding the exponents.
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Step 4.19.4.1.11.1
Move .
Step 4.19.4.1.11.2
Multiply by .
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Step 4.19.4.1.11.2.1
Raise to the power of .
Step 4.19.4.1.11.2.2
Use the power rule to combine exponents.
Step 4.19.4.1.11.3
Write as a fraction with a common denominator.
Step 4.19.4.1.11.4
Combine the numerators over the common denominator.
Step 4.19.4.1.11.5
Add and .
Step 4.19.4.1.12
Move to the left of .
Step 4.19.4.1.13
Multiply by .
Step 4.19.4.1.14
Combine and .
Step 4.19.4.1.15
Move the negative in front of the fraction.
Step 4.19.4.2
To write as a fraction with a common denominator, multiply by .
Step 4.19.4.3
Combine and .
Step 4.19.4.4
Combine the numerators over the common denominator.
Step 4.19.4.5
Simplify each term.
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Step 4.19.4.5.1
Simplify the numerator.
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Step 4.19.4.5.1.1
Factor out of .
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Step 4.19.4.5.1.1.1
Move .
Step 4.19.4.5.1.1.2
Factor out of .
Step 4.19.4.5.1.1.3
Factor out of .
Step 4.19.4.5.1.1.4
Factor out of .
Step 4.19.4.5.1.2
Multiply by .
Step 4.19.4.5.1.3
Subtract from .
Step 4.19.4.5.2
Move to the left of .
Step 4.19.4.6
Subtract from .
Step 4.19.5
Combine terms.
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Step 4.19.5.1
Multiply by .
Step 4.19.5.2
Combine.
Step 4.19.5.3
Apply the distributive property.
Step 4.19.5.4
Cancel the common factor of .
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Step 4.19.5.4.1
Cancel the common factor.
Step 4.19.5.4.2
Rewrite the expression.
Step 4.19.5.5
Multiply by .
Step 4.19.5.6
Multiply by .
Step 4.19.5.7
Combine and .
Step 4.19.5.8
Multiply by .
Step 4.19.5.9
Factor out of .
Step 4.19.5.10
Cancel the common factors.
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Step 4.19.5.10.1
Factor out of .
Step 4.19.5.10.2
Cancel the common factor.
Step 4.19.5.10.3
Rewrite the expression.
Step 4.19.5.11
Move the negative in front of the fraction.
Step 4.19.6
Simplify the numerator.
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Step 4.19.6.1
To write as a fraction with a common denominator, multiply by .
Step 4.19.6.2
Combine the numerators over the common denominator.
Step 4.19.6.3
Simplify the numerator.
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Step 4.19.6.3.1
Multiply by by adding the exponents.
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Step 4.19.6.3.1.1
Move .
Step 4.19.6.3.1.2
Use the power rule to combine exponents.
Step 4.19.6.3.1.3
Combine the numerators over the common denominator.
Step 4.19.6.3.1.4
Add and .
Step 4.19.6.3.1.5
Divide by .
Step 4.19.6.3.2
Simplify .
Step 4.19.6.4
To write as a fraction with a common denominator, multiply by .
Step 4.19.6.5
Combine the numerators over the common denominator.
Step 4.19.6.6
Simplify the numerator.
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Step 4.19.6.6.1
Multiply by by adding the exponents.
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Step 4.19.6.6.1.1
Move .
Step 4.19.6.6.1.2
Use the power rule to combine exponents.
Step 4.19.6.6.1.3
Combine the numerators over the common denominator.
Step 4.19.6.6.1.4
Add and .
Step 4.19.6.6.1.5
Divide by .
Step 4.19.6.6.2
Factor by grouping.
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Step 4.19.6.6.2.1
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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Step 4.19.6.6.2.1.1
Factor out of .
Step 4.19.6.6.2.1.2
Rewrite as plus
Step 4.19.6.6.2.1.3
Apply the distributive property.
Step 4.19.6.6.2.2
Factor out the greatest common factor from each group.
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Step 4.19.6.6.2.2.1
Group the first two terms and the last two terms.
Step 4.19.6.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 4.19.6.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 4.19.7
Multiply the numerator by the reciprocal of the denominator.
Step 4.19.8
Combine.
Step 4.19.9
Multiply by by adding the exponents.
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Step 4.19.9.1
Move .
Step 4.19.9.2
Multiply by .
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Step 4.19.9.2.1
Raise to the power of .
Step 4.19.9.2.2
Use the power rule to combine exponents.
Step 4.19.9.3
Write as a fraction with a common denominator.
Step 4.19.9.4
Combine the numerators over the common denominator.
Step 4.19.9.5
Add and .
Step 4.19.10
Multiply by .
Step 4.19.11
Move to the left of .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .