Calculus Examples

Find dy/dx y=((x-7)(x^2+4x))/(x^3)
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2
Multiply the exponents in .
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Step 3.2.1
Apply the power rule and multiply exponents, .
Step 3.2.2
Multiply by .
Step 3.3
Differentiate using the Product Rule which states that is where and .
Step 3.4
Differentiate.
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Step 3.4.1
By the Sum Rule, the derivative of with respect to is .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.4
Differentiate using the Power Rule which states that is where .
Step 3.4.5
Multiply by .
Step 3.4.6
By the Sum Rule, the derivative of with respect to is .
Step 3.4.7
Differentiate using the Power Rule which states that is where .
Step 3.4.8
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.9
Simplify the expression.
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Step 3.4.9.1
Add and .
Step 3.4.9.2
Multiply by .
Step 3.4.10
Differentiate using the Power Rule which states that is where .
Step 3.4.11
Simplify with factoring out.
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Step 3.4.11.1
Multiply by .
Step 3.4.11.2
Factor out of .
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Step 3.4.11.2.1
Factor out of .
Step 3.4.11.2.2
Factor out of .
Step 3.4.11.2.3
Factor out of .
Step 3.5
Cancel the common factors.
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Step 3.5.1
Factor out of .
Step 3.5.2
Cancel the common factor.
Step 3.5.3
Rewrite the expression.
Step 3.6
Factor out of .
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Step 3.6.1
Factor out of .
Step 3.6.2
Factor out of .
Step 3.7
Cancel the common factors.
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Step 3.7.1
Factor out of .
Step 3.7.2
Cancel the common factor.
Step 3.7.3
Rewrite the expression.
Step 3.8
Simplify.
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Step 3.8.1
Rewrite the expression using the negative exponent rule .
Step 3.8.2
Apply the distributive property.
Step 3.8.3
Simplify the numerator.
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Step 3.8.3.1
Simplify each term.
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Step 3.8.3.1.1
Expand using the FOIL Method.
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Step 3.8.3.1.1.1
Apply the distributive property.
Step 3.8.3.1.1.2
Apply the distributive property.
Step 3.8.3.1.1.3
Apply the distributive property.
Step 3.8.3.1.2
Simplify and combine like terms.
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Step 3.8.3.1.2.1
Simplify each term.
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Step 3.8.3.1.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.8.3.1.2.1.2
Multiply by by adding the exponents.
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Step 3.8.3.1.2.1.2.1
Move .
Step 3.8.3.1.2.1.2.2
Multiply by .
Step 3.8.3.1.2.1.3
Move to the left of .
Step 3.8.3.1.2.1.4
Multiply by .
Step 3.8.3.1.2.1.5
Multiply by .
Step 3.8.3.1.2.2
Subtract from .
Step 3.8.3.1.3
Multiply by .
Step 3.8.3.1.4
Combine and .
Step 3.8.3.1.5
Expand using the FOIL Method.
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Step 3.8.3.1.5.1
Apply the distributive property.
Step 3.8.3.1.5.2
Apply the distributive property.
Step 3.8.3.1.5.3
Apply the distributive property.
Step 3.8.3.1.6
Simplify and combine like terms.
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Step 3.8.3.1.6.1
Simplify each term.
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Step 3.8.3.1.6.1.1
Multiply by .
Step 3.8.3.1.6.1.2
Cancel the common factor of .
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Step 3.8.3.1.6.1.2.1
Factor out of .
Step 3.8.3.1.6.1.2.2
Cancel the common factor.
Step 3.8.3.1.6.1.2.3
Rewrite the expression.
Step 3.8.3.1.6.1.3
Multiply by .
Step 3.8.3.1.6.1.4
Multiply by .
Step 3.8.3.1.6.1.5
Multiply .
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Step 3.8.3.1.6.1.5.1
Combine and .
Step 3.8.3.1.6.1.5.2
Multiply by .
Step 3.8.3.1.6.2
Add and .
Step 3.8.3.1.7
Apply the distributive property.
Step 3.8.3.1.8
Simplify.
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Step 3.8.3.1.8.1
Multiply by by adding the exponents.
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Step 3.8.3.1.8.1.1
Move .
Step 3.8.3.1.8.1.2
Multiply by .
Step 3.8.3.1.8.2
Cancel the common factor of .
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Step 3.8.3.1.8.2.1
Cancel the common factor.
Step 3.8.3.1.8.2.2
Rewrite the expression.
Step 3.8.3.2
Add and .
Step 3.8.3.3
Combine the opposite terms in .
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Step 3.8.3.3.1
Subtract from .
Step 3.8.3.3.2
Add and .
Step 3.8.3.4
Add and .
Step 3.8.3.5
Add and .
Step 3.8.3.6
Add and .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .