Calculus Examples

Find dR/dM R=M^2(C/2-M/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 Product Rule which states that is where and .
Step 3.2
Differentiate.
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Step 3.2.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.3
Add and .
Step 3.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.5
Combine and .
Step 3.2.6
Differentiate using the Power Rule which states that is where .
Step 3.2.7
Multiply by .
Step 3.2.8
Differentiate using the Power Rule which states that is where .
Step 3.2.9
Move to the left of .
Step 3.3
Combine and using a common denominator.
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Step 3.3.1
Reorder and .
Step 3.3.2
To write as a fraction with a common denominator, multiply by .
Step 3.3.3
Combine and .
Step 3.3.4
Combine the numerators over the common denominator.
Step 3.4
Multiply by .
Step 3.5
Simplify.
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Step 3.5.1
Apply the distributive property.
Step 3.5.2
Simplify the numerator.
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Step 3.5.2.1
Simplify each term.
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Step 3.5.2.1.1
Cancel the common factor of .
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Step 3.5.2.1.1.1
Factor out of .
Step 3.5.2.1.1.2
Cancel the common factor.
Step 3.5.2.1.1.3
Rewrite the expression.
Step 3.5.2.1.2
Cancel the common factor of .
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Step 3.5.2.1.2.1
Move the leading negative in into the numerator.
Step 3.5.2.1.2.2
Factor out of .
Step 3.5.2.1.2.3
Cancel the common factor.
Step 3.5.2.1.2.4
Rewrite the expression.
Step 3.5.2.1.3
Multiply by .
Step 3.5.2.1.4
Raise to the power of .
Step 3.5.2.1.5
Raise to the power of .
Step 3.5.2.1.6
Use the power rule to combine exponents.
Step 3.5.2.1.7
Add and .
Step 3.5.2.2
Subtract from .
Step 3.5.3
Reorder terms.
Step 3.5.4
Factor out of .
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Step 3.5.4.1
Factor out of .
Step 3.5.4.2
Factor out of .
Step 3.5.4.3
Factor out of .
Step 3.5.5
Cancel the common factor of .
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Step 3.5.5.1
Cancel the common factor.
Step 3.5.5.2
Divide by .
Step 3.5.6
Apply the distributive property.
Step 3.5.7
Rewrite using the commutative property of multiplication.
Step 3.5.8
Multiply by by adding the exponents.
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Step 3.5.8.1
Move .
Step 3.5.8.2
Multiply by .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .