Calculus Examples

Find dy/dt y=((t^2)/(t^3-4t))^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 chain rule, which states that is where and .
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Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.3
Differentiate.
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Step 3.3.1
Differentiate using the Power Rule which states that is where .
Step 3.3.2
Move to the left of .
Step 3.3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.3.4
Differentiate using the Power Rule which states that is where .
Step 3.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.6
Differentiate using the Power Rule which states that is where .
Step 3.3.7
Combine fractions.
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Step 3.3.7.1
Multiply by .
Step 3.3.7.2
Combine and .
Step 3.3.7.3
Move to the left of .
Step 3.4
Simplify.
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Step 3.4.1
Apply the product rule to .
Step 3.4.2
Apply the distributive property.
Step 3.4.3
Apply the distributive property.
Step 3.4.4
Apply the distributive property.
Step 3.4.5
Apply the distributive property.
Step 3.4.6
Combine terms.
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Step 3.4.6.1
Raise to the power of .
Step 3.4.6.2
Use the power rule to combine exponents.
Step 3.4.6.3
Add and .
Step 3.4.6.4
Multiply by .
Step 3.4.6.5
Multiply by .
Step 3.4.6.6
Raise to the power of .
Step 3.4.6.7
Raise to the power of .
Step 3.4.6.8
Use the power rule to combine exponents.
Step 3.4.6.9
Add and .
Step 3.4.6.10
Multiply by .
Step 3.4.6.11
Multiply by .
Step 3.4.6.12
Multiply by by adding the exponents.
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Step 3.4.6.12.1
Move .
Step 3.4.6.12.2
Use the power rule to combine exponents.
Step 3.4.6.12.3
Add and .
Step 3.4.6.13
Multiply by .
Step 3.4.6.14
Multiply by .
Step 3.4.6.15
Multiply by .
Step 3.4.6.16
Subtract from .
Step 3.4.6.17
Add and .
Step 3.4.6.18
Multiply the exponents in .
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Step 3.4.6.18.1
Apply the power rule and multiply exponents, .
Step 3.4.6.18.2
Multiply by .
Step 3.4.6.19
Multiply by .
Step 3.4.6.20
Multiply by by adding the exponents.
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Step 3.4.6.20.1
Use the power rule to combine exponents.
Step 3.4.6.20.2
Add and .
Step 3.4.7
Reorder terms.
Step 3.4.8
Simplify the numerator.
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Step 3.4.8.1
Factor out of .
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Step 3.4.8.1.1
Factor out of .
Step 3.4.8.1.2
Factor out of .
Step 3.4.8.1.3
Factor out of .
Step 3.4.8.2
Multiply by by adding the exponents.
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Step 3.4.8.2.1
Move .
Step 3.4.8.2.2
Use the power rule to combine exponents.
Step 3.4.8.2.3
Add and .
Step 3.4.9
Simplify the denominator.
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Step 3.4.9.1
Factor out of .
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Step 3.4.9.1.1
Factor out of .
Step 3.4.9.1.2
Factor out of .
Step 3.4.9.1.3
Factor out of .
Step 3.4.9.2
Rewrite as .
Step 3.4.9.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.4.9.4
Apply the product rule to .
Step 3.4.9.5
Apply the distributive property.
Step 3.4.9.6
Multiply by .
Step 3.4.9.7
Move to the left of .
Step 3.4.9.8
Use the Binomial Theorem.
Step 3.4.9.9
Simplify each term.
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Step 3.4.9.9.1
Multiply the exponents in .
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Step 3.4.9.9.1.1
Apply the power rule and multiply exponents, .
Step 3.4.9.9.1.2
Multiply by .
Step 3.4.9.9.2
Rewrite using the commutative property of multiplication.
Step 3.4.9.9.3
Multiply by .
Step 3.4.9.9.4
Multiply the exponents in .
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Step 3.4.9.9.4.1
Apply the power rule and multiply exponents, .
Step 3.4.9.9.4.2
Multiply by .
Step 3.4.9.9.5
Multiply by by adding the exponents.
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Step 3.4.9.9.5.1
Move .
Step 3.4.9.9.5.2
Multiply by .
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Step 3.4.9.9.5.2.1
Raise to the power of .
Step 3.4.9.9.5.2.2
Use the power rule to combine exponents.
Step 3.4.9.9.5.3
Add and .
Step 3.4.9.9.6
Multiply the exponents in .
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Step 3.4.9.9.6.1
Apply the power rule and multiply exponents, .
Step 3.4.9.9.6.2
Multiply by .
Step 3.4.9.9.7
Apply the product rule to .
Step 3.4.9.9.8
Rewrite using the commutative property of multiplication.
Step 3.4.9.9.9
Multiply by by adding the exponents.
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Step 3.4.9.9.9.1
Move .
Step 3.4.9.9.9.2
Use the power rule to combine exponents.
Step 3.4.9.9.9.3
Add and .
Step 3.4.9.9.10
Raise to the power of .
Step 3.4.9.9.11
Multiply by .
Step 3.4.9.9.12
Apply the product rule to .
Step 3.4.9.9.13
Rewrite using the commutative property of multiplication.
Step 3.4.9.9.14
Multiply by by adding the exponents.
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Step 3.4.9.9.14.1
Move .
Step 3.4.9.9.14.2
Use the power rule to combine exponents.
Step 3.4.9.9.14.3
Add and .
Step 3.4.9.9.15
Raise to the power of .
Step 3.4.9.9.16
Multiply by .
Step 3.4.9.9.17
Apply the product rule to .
Step 3.4.9.9.18
Raise to the power of .
Step 3.4.9.10
Factor out of .
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Step 3.4.9.10.1
Factor out of .
Step 3.4.9.10.2
Factor out of .
Step 3.4.9.10.3
Factor out of .
Step 3.4.9.10.4
Factor out of .
Step 3.4.9.10.5
Factor out of .
Step 3.4.9.10.6
Factor out of .
Step 3.4.9.10.7
Factor out of .
Step 3.4.9.10.8
Factor out of .
Step 3.4.9.10.9
Factor out of .
Step 3.4.9.11
Make each term match the terms from the binomial theorem formula.
Step 3.4.9.12
Factor using the binomial theorem.
Step 3.4.10
Cancel the common factor of and .
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Step 3.4.10.1
Factor out of .
Step 3.4.10.2
Cancel the common factors.
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Step 3.4.10.2.1
Factor out of .
Step 3.4.10.2.2
Cancel the common factor.
Step 3.4.10.2.3
Rewrite the expression.
Step 3.4.11
Move to the left of .
Step 3.4.12
Factor out of .
Step 3.4.13
Rewrite as .
Step 3.4.14
Factor out of .
Step 3.4.15
Rewrite as .
Step 3.4.16
Move the negative in front of the fraction.
Step 3.4.17
Reorder factors in .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .