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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Evaluate .
Step 2.3.1
Differentiate using the Product Rule which states that is where and .
Step 2.3.2
Differentiate using the chain rule, which states that is where and .
Step 2.3.2.1
To apply the Chain Rule, set as .
Step 2.3.2.2
Differentiate using the Power Rule which states that is where .
Step 2.3.2.3
Replace all occurrences of with .
Step 2.3.3
Rewrite as .
Step 2.3.4
Differentiate using the Power Rule which states that is where .
Step 2.3.5
Move to the left of .
Step 2.3.6
Multiply by .
Step 2.4
Reorder terms.
Step 3
Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the Product Rule which states that is where and .
Step 3.3
Differentiate using the chain rule, which states that is where and .
Step 3.3.1
To apply the Chain Rule, set as .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Replace all occurrences of with .
Step 3.4
Move to the left of .
Step 3.5
Rewrite as .
Step 3.6
Differentiate using the Power Rule which states that is where .
Step 3.7
Move to the left of .
Step 3.8
Simplify.
Step 3.8.1
Apply the distributive property.
Step 3.8.2
Combine terms.
Step 3.8.2.1
Multiply by .
Step 3.8.2.2
Multiply by .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Step 5.1
Subtract from both sides of the equation.
Step 5.2
Move all terms not containing to the right side of the equation.
Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Subtract from both sides of the equation.
Step 5.3
Factor out of .
Step 5.3.1
Factor out of .
Step 5.3.2
Factor out of .
Step 5.3.3
Factor out of .
Step 5.4
Divide each term in by and simplify.
Step 5.4.1
Divide each term in by .
Step 5.4.2
Simplify the left side.
Step 5.4.2.1
Cancel the common factor of .
Step 5.4.2.1.1
Cancel the common factor.
Step 5.4.2.1.2
Rewrite the expression.
Step 5.4.2.2
Cancel the common factor of .
Step 5.4.2.2.1
Cancel the common factor.
Step 5.4.2.2.2
Rewrite the expression.
Step 5.4.2.3
Cancel the common factor of .
Step 5.4.2.3.1
Cancel the common factor.
Step 5.4.2.3.2
Divide by .
Step 5.4.3
Simplify the right side.
Step 5.4.3.1
Simplify each term.
Step 5.4.3.1.1
Cancel the common factor of and .
Step 5.4.3.1.1.1
Factor out of .
Step 5.4.3.1.1.2
Cancel the common factors.
Step 5.4.3.1.1.2.1
Factor out of .
Step 5.4.3.1.1.2.2
Cancel the common factor.
Step 5.4.3.1.1.2.3
Rewrite the expression.
Step 5.4.3.1.2
Cancel the common factor of and .
Step 5.4.3.1.2.1
Factor out of .
Step 5.4.3.1.2.2
Cancel the common factors.
Step 5.4.3.1.2.2.1
Cancel the common factor.
Step 5.4.3.1.2.2.2
Rewrite the expression.
Step 5.4.3.1.3
Cancel the common factor of and .
Step 5.4.3.1.3.1
Factor out of .
Step 5.4.3.1.3.2
Cancel the common factors.
Step 5.4.3.1.3.2.1
Factor out of .
Step 5.4.3.1.3.2.2
Cancel the common factor.
Step 5.4.3.1.3.2.3
Rewrite the expression.
Step 5.4.3.1.4
Move the negative in front of the fraction.
Step 5.4.3.1.5
Cancel the common factor of and .
Step 5.4.3.1.5.1
Factor out of .
Step 5.4.3.1.5.2
Cancel the common factors.
Step 5.4.3.1.5.2.1
Factor out of .
Step 5.4.3.1.5.2.2
Cancel the common factor.
Step 5.4.3.1.5.2.3
Rewrite the expression.
Step 5.4.3.1.6
Move the negative in front of the fraction.
Step 5.4.3.2
To write as a fraction with a common denominator, multiply by .
Step 5.4.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 5.4.3.3.1
Multiply by .
Step 5.4.3.3.2
Reorder the factors of .
Step 5.4.3.4
Combine the numerators over the common denominator.
Step 5.4.3.5
Simplify the numerator.
Step 5.4.3.5.1
Factor out of .
Step 5.4.3.5.1.1
Factor out of .
Step 5.4.3.5.1.2
Factor out of .
Step 5.4.3.5.1.3
Factor out of .
Step 5.4.3.5.2
Multiply by by adding the exponents.
Step 5.4.3.5.2.1
Multiply by .
Step 5.4.3.5.2.1.1
Raise to the power of .
Step 5.4.3.5.2.1.2
Use the power rule to combine exponents.
Step 5.4.3.5.2.2
Add and .
Step 5.4.3.5.3
Rewrite as .
Step 5.4.3.5.4
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 5.4.3.5.5
Simplify.
Step 5.4.3.5.5.1
Multiply by .
Step 5.4.3.5.5.2
One to any power is one.
Step 5.4.3.6
To write as a fraction with a common denominator, multiply by .
Step 5.4.3.7
To write as a fraction with a common denominator, multiply by .
Step 5.4.3.8
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 5.4.3.8.1
Multiply by .
Step 5.4.3.8.2
Multiply by .
Step 5.4.3.8.3
Reorder the factors of .
Step 5.4.3.8.4
Reorder the factors of .
Step 5.4.3.9
Combine the numerators over the common denominator.
Step 5.4.3.10
Simplify the numerator.
Step 5.4.3.10.1
Multiply by by adding the exponents.
Step 5.4.3.10.1.1
Move .
Step 5.4.3.10.1.2
Multiply by .
Step 5.4.3.10.2
Apply the distributive property.
Step 5.4.3.10.3
Multiply by .
Step 5.4.3.10.4
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.4.3.10.5
Simplify each term.
Step 5.4.3.10.5.1
Multiply by by adding the exponents.
Step 5.4.3.10.5.1.1
Move .
Step 5.4.3.10.5.1.2
Multiply by .
Step 5.4.3.10.5.1.2.1
Raise to the power of .
Step 5.4.3.10.5.1.2.2
Use the power rule to combine exponents.
Step 5.4.3.10.5.1.3
Add and .
Step 5.4.3.10.5.2
Multiply by by adding the exponents.
Step 5.4.3.10.5.2.1
Move .
Step 5.4.3.10.5.2.2
Multiply by .
Step 5.4.3.10.5.3
Multiply by .
Step 5.4.3.10.5.4
Multiply by .
Step 5.4.3.10.6
Combine the opposite terms in .
Step 5.4.3.10.6.1
Subtract from .
Step 5.4.3.10.6.2
Add and .
Step 5.4.3.10.6.3
Subtract from .
Step 5.4.3.10.6.4
Add and .
Step 5.4.3.10.7
Multiply by by adding the exponents.
Step 5.4.3.10.7.1
Move .
Step 5.4.3.10.7.2
Multiply by .
Step 6
Replace with .