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Calculus Examples
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
Step 3.1
Differentiate using the Product Rule which states that is where and .
Step 3.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
To write as a fraction with a common denominator, multiply by .
Step 3.4
Combine and .
Step 3.5
Combine the numerators over the common denominator.
Step 3.6
Simplify the numerator.
Step 3.6.1
Multiply by .
Step 3.6.2
Subtract from .
Step 3.7
Combine fractions.
Step 3.7.1
Move the negative in front of the fraction.
Step 3.7.2
Combine and .
Step 3.7.3
Move to the denominator using the negative exponent rule .
Step 3.7.4
Combine and .
Step 3.8
By the Sum Rule, the derivative of with respect to is .
Step 3.9
Rewrite as .
Step 3.10
Since is constant with respect to , the derivative of with respect to is .
Step 3.11
Combine fractions.
Step 3.11.1
Add and .
Step 3.11.2
Combine and .
Step 3.12
Differentiate using the Power Rule which states that is where .
Step 3.13
Multiply by .
Step 3.14
To write as a fraction with a common denominator, multiply by .
Step 3.15
Combine and .
Step 3.16
Combine the numerators over the common denominator.
Step 3.17
Multiply by by adding the exponents.
Step 3.17.1
Move .
Step 3.17.2
Use the power rule to combine exponents.
Step 3.17.3
Combine the numerators over the common denominator.
Step 3.17.4
Add and .
Step 3.17.5
Divide by .
Step 3.18
Simplify .
Step 3.19
Move to the left of .
Step 3.20
Simplify.
Step 3.20.1
Apply the distributive property.
Step 3.20.2
Multiply by .
Step 4
Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Evaluate .
Step 4.2.1
Differentiate using the Product Rule which states that is where and .
Step 4.2.2
Rewrite as .
Step 4.2.3
Differentiate using the Power Rule which states that is where .
Step 4.2.4
Multiply by .
Step 4.3
Differentiate using the Constant Rule.
Step 4.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.2
Add and .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Step 6.1
Multiply both sides by .
Step 6.2
Simplify.
Step 6.2.1
Simplify the left side.
Step 6.2.1.1
Simplify .
Step 6.2.1.1.1
Rewrite using the commutative property of multiplication.
Step 6.2.1.1.2
Cancel the common factor of .
Step 6.2.1.1.2.1
Cancel the common factor.
Step 6.2.1.1.2.2
Rewrite the expression.
Step 6.2.1.1.3
Cancel the common factor of .
Step 6.2.1.1.3.1
Cancel the common factor.
Step 6.2.1.1.3.2
Rewrite the expression.
Step 6.2.1.1.4
Reorder and .
Step 6.2.2
Simplify the right side.
Step 6.2.2.1
Simplify .
Step 6.2.2.1.1
Apply the distributive property.
Step 6.2.2.1.2
Reorder.
Step 6.2.2.1.2.1
Rewrite using the commutative property of multiplication.
Step 6.2.2.1.2.2
Rewrite using the commutative property of multiplication.
Step 6.2.2.1.2.3
Move .
Step 6.3
Solve for .
Step 6.3.1
Subtract from both sides of the equation.
Step 6.3.2
Move all terms not containing to the right side of the equation.
Step 6.3.2.1
Subtract from both sides of the equation.
Step 6.3.2.2
Subtract from both sides of the equation.
Step 6.3.3
Factor out of .
Step 6.3.3.1
Factor out of .
Step 6.3.3.2
Factor out of .
Step 6.3.3.3
Factor out of .
Step 6.3.4
Divide each term in by and simplify.
Step 6.3.4.1
Divide each term in by .
Step 6.3.4.2
Simplify the left side.
Step 6.3.4.2.1
Cancel the common factor.
Step 6.3.4.2.2
Rewrite the expression.
Step 6.3.4.2.3
Cancel the common factor.
Step 6.3.4.2.4
Divide by .
Step 6.3.4.3
Simplify the right side.
Step 6.3.4.3.1
Simplify terms.
Step 6.3.4.3.1.1
Simplify each term.
Step 6.3.4.3.1.1.1
Move the negative in front of the fraction.
Step 6.3.4.3.1.1.2
Move the negative in front of the fraction.
Step 6.3.4.3.1.2
Simplify terms.
Step 6.3.4.3.1.2.1
Combine the numerators over the common denominator.
Step 6.3.4.3.1.2.2
Factor out of .
Step 6.3.4.3.1.2.2.1
Factor out of .
Step 6.3.4.3.1.2.2.2
Factor out of .
Step 6.3.4.3.1.2.2.3
Factor out of .
Step 6.3.4.3.1.2.3
Combine the numerators over the common denominator.
Step 6.3.4.3.2
Simplify the numerator.
Step 6.3.4.3.2.1
Factor out of .
Step 6.3.4.3.2.1.1
Factor out of .
Step 6.3.4.3.2.1.2
Factor out of .
Step 6.3.4.3.2.1.3
Factor out of .
Step 6.3.4.3.2.2
Apply the distributive property.
Step 6.3.4.3.2.3
Move to the left of .
Step 6.3.4.3.2.4
Rewrite as .
Step 7
Replace with .