Calculus Examples

Find dy/dx y=(6x-5)^2(3-x^5)^2
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
Rewrite as .
Step 3.2
Expand using the FOIL Method.
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Step 3.2.1
Apply the distributive property.
Step 3.2.2
Apply the distributive property.
Step 3.2.3
Apply the distributive property.
Step 3.3
Simplify and combine like terms.
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Step 3.3.1
Simplify each term.
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Step 3.3.1.1
Rewrite using the commutative property of multiplication.
Step 3.3.1.2
Multiply by by adding the exponents.
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Step 3.3.1.2.1
Move .
Step 3.3.1.2.2
Multiply by .
Step 3.3.1.3
Multiply by .
Step 3.3.1.4
Multiply by .
Step 3.3.1.5
Multiply by .
Step 3.3.1.6
Multiply by .
Step 3.3.2
Subtract from .
Step 3.4
Differentiate using the Product Rule which states that is where and .
Step 3.5
Differentiate using the chain rule, which states that is where and .
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Step 3.5.1
To apply the Chain Rule, set as .
Step 3.5.2
Differentiate using the Power Rule which states that is where .
Step 3.5.3
Replace all occurrences of with .
Step 3.6
Differentiate.
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Step 3.6.1
Move to the left of .
Step 3.6.2
By the Sum Rule, the derivative of with respect to is .
Step 3.6.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.4
Add and .
Step 3.6.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.6
Multiply by .
Step 3.6.7
Differentiate using the Power Rule which states that is where .
Step 3.6.8
Multiply by .
Step 3.6.9
By the Sum Rule, the derivative of with respect to is .
Step 3.6.10
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.11
Differentiate using the Power Rule which states that is where .
Step 3.6.12
Multiply by .
Step 3.6.13
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.14
Differentiate using the Power Rule which states that is where .
Step 3.6.15
Multiply by .
Step 3.6.16
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.17
Add and .
Step 3.7
Simplify.
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Step 3.7.1
Apply the distributive property.
Step 3.7.2
Multiply by .
Step 3.7.3
Multiply by .
Step 3.7.4
Multiply by .
Step 3.7.5
Factor out of .
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Step 3.7.5.1
Factor out of .
Step 3.7.5.2
Factor out of .
Step 3.7.5.3
Factor out of .
Step 3.7.6
Reorder the factors of .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .