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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
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
Step 3.2.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Rewrite as .
Step 3.2.4
Multiply by .
Step 3.3
Evaluate .
Step 3.3.1
Differentiate using the chain rule, which states that is where and .
Step 3.3.1.1
To apply the Chain Rule, set as .
Step 3.3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.3.1.3
Replace all occurrences of with .
Step 3.3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3.3
Differentiate using the Power Rule which states that is where .
Step 3.3.4
Rewrite as .
Step 3.3.5
To write as a fraction with a common denominator, multiply by .
Step 3.3.6
Combine and .
Step 3.3.7
Combine the numerators over the common denominator.
Step 3.3.8
Simplify the numerator.
Step 3.3.8.1
Multiply by .
Step 3.3.8.2
Subtract from .
Step 3.3.9
Move the negative in front of the fraction.
Step 3.3.10
Combine and .
Step 3.3.11
Move to the denominator using the negative exponent rule .
Step 3.4
Simplify.
Step 3.4.1
Combine terms.
Step 3.4.1.1
To write as a fraction with a common denominator, multiply by .
Step 3.4.1.2
Combine and .
Step 3.4.1.3
Combine the numerators over the common denominator.
Step 3.4.1.4
Combine and .
Step 3.4.2
Reorder terms.
Step 4
Step 4.1
Differentiate using the chain rule, which states that is where and .
Step 4.1.1
To apply the Chain Rule, set as .
Step 4.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.1.3
Replace all occurrences of with .
Step 4.2
Differentiate using the Product Rule which states that is where and .
Step 4.3
Rewrite as .
Step 4.4
Differentiate using the Power Rule which states that is where .
Step 4.5
Multiply by .
Step 4.6
Simplify.
Step 4.6.1
Apply the distributive property.
Step 4.6.2
Reorder terms.
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
Cancel the common factor of .
Step 6.2.1.1.1.1
Cancel the common factor.
Step 6.2.1.1.1.2
Rewrite the expression.
Step 6.2.1.1.2
Multiply by .
Step 6.2.1.1.3
To write as a fraction with a common denominator, multiply by .
Step 6.2.1.1.4
Simplify terms.
Step 6.2.1.1.4.1
Combine and .
Step 6.2.1.1.4.2
Combine the numerators over the common denominator.
Step 6.2.1.1.5
Simplify the numerator.
Step 6.2.1.1.5.1
Multiply by .
Step 6.2.1.1.5.2
Apply the distributive property.
Step 6.2.1.1.5.3
Multiply by .
Step 6.2.1.1.6
To write as a fraction with a common denominator, multiply by .
Step 6.2.1.1.7
Combine and .
Step 6.2.1.1.8
Combine the numerators over the common denominator.
Step 6.2.1.1.9
Rewrite using the commutative property of multiplication.
Step 6.2.1.1.10
Factor out of .
Step 6.2.1.1.11
Factor out of .
Step 6.2.1.1.12
Factor out of .
Step 6.2.1.1.13
Factor out of .
Step 6.2.1.1.14
Factor out of .
Step 6.2.1.1.15
Factor out of .
Step 6.2.1.1.16
Factor out of .
Step 6.2.1.1.17
Simplify the expression.
Step 6.2.1.1.17.1
Rewrite as .
Step 6.2.1.1.17.2
Move the negative in front of the fraction.
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
Multiply by by adding the exponents.
Step 6.2.2.1.2.1
Move .
Step 6.2.2.1.2.2
Multiply by .
Step 6.2.2.1.2.2.1
Raise to the power of .
Step 6.2.2.1.2.2.2
Use the power rule to combine exponents.
Step 6.2.2.1.2.3
Add and .
Step 6.2.2.1.3
Reorder factors in .
Step 6.2.2.1.4
Reorder and .
Step 6.3
Solve for .
Step 6.3.1
Multiply both sides by .
Step 6.3.2
Simplify.
Step 6.3.2.1
Simplify the left side.
Step 6.3.2.1.1
Simplify .
Step 6.3.2.1.1.1
Cancel the common factor of .
Step 6.3.2.1.1.1.1
Move the leading negative in into the numerator.
Step 6.3.2.1.1.1.2
Cancel the common factor.
Step 6.3.2.1.1.1.3
Rewrite the expression.
Step 6.3.2.1.1.2
Apply the distributive property.
Step 6.3.2.1.1.3
Simplify.
Step 6.3.2.1.1.3.1
Multiply by .
Step 6.3.2.1.1.3.2
Multiply .
Step 6.3.2.1.1.3.2.1
Multiply by .
Step 6.3.2.1.1.3.2.2
Multiply by .
Step 6.3.2.1.1.3.3
Multiply .
Step 6.3.2.1.1.3.3.1
Multiply by .
Step 6.3.2.1.1.3.3.2
Multiply by .
Step 6.3.2.1.1.3.4
Multiply by .
Step 6.3.2.1.1.4
Remove parentheses.
Step 6.3.2.1.1.5
Simplify the expression.
Step 6.3.2.1.1.5.1
Move .
Step 6.3.2.1.1.5.2
Move .
Step 6.3.2.1.1.5.3
Reorder and .
Step 6.3.2.2
Simplify the right side.
Step 6.3.2.2.1
Simplify .
Step 6.3.2.2.1.1
Apply the distributive property.
Step 6.3.2.2.1.2
Reorder.
Step 6.3.2.2.1.2.1
Rewrite using the commutative property of multiplication.
Step 6.3.2.2.1.2.2
Rewrite using the commutative property of multiplication.
Step 6.3.3
Solve for .
Step 6.3.3.1
Subtract from both sides of the equation.
Step 6.3.3.2
Move all terms not containing to the right side of the equation.
Step 6.3.3.2.1
Subtract from both sides of the equation.
Step 6.3.3.2.2
Subtract from both sides of the equation.
Step 6.3.3.3
Factor out of .
Step 6.3.3.3.1
Factor out of .
Step 6.3.3.3.2
Factor out of .
Step 6.3.3.3.3
Factor out of .
Step 6.3.3.3.4
Factor out of .
Step 6.3.3.3.5
Factor out of .
Step 6.3.3.4
Divide each term in by and simplify.
Step 6.3.3.4.1
Divide each term in by .
Step 6.3.3.4.2
Simplify the left side.
Step 6.3.3.4.2.1
Cancel the common factor.
Step 6.3.3.4.2.2
Divide by .
Step 6.3.3.4.3
Simplify the right side.
Step 6.3.3.4.3.1
Simplify each term.
Step 6.3.3.4.3.1.1
Move the negative in front of the fraction.
Step 6.3.3.4.3.1.2
Move the negative in front of the fraction.
Step 6.3.3.4.3.2
Simplify terms.
Step 6.3.3.4.3.2.1
Combine the numerators over the common denominator.
Step 6.3.3.4.3.2.2
Combine the numerators over the common denominator.
Step 6.3.3.4.3.2.3
Factor out of .
Step 6.3.3.4.3.2.3.1
Factor out of .
Step 6.3.3.4.3.2.3.2
Factor out of .
Step 6.3.3.4.3.2.3.3
Factor out of .
Step 6.3.3.4.3.2.3.4
Factor out of .
Step 6.3.3.4.3.2.3.5
Factor out of .
Step 7
Replace with .