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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
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
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate.
Step 3.3.1
Combine and .
Step 3.3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.4
Add and .
Step 3.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.6
Combine fractions.
Step 3.3.6.1
Combine and .
Step 3.3.6.2
Move the negative in front of the fraction.
Step 3.3.7
Differentiate using the Power Rule which states that is where .
Step 3.3.8
Multiply by .
Step 3.3.9
Differentiate using the Power Rule which states that is where .
Step 3.3.10
Move .
Step 3.4
To write as a fraction with a common denominator, multiply by .
Step 3.5
Combine and .
Step 3.6
Combine the numerators over the common denominator.
Step 3.7
Simplify.
Step 3.7.1
Apply the distributive property.
Step 3.7.2
Apply the distributive property.
Step 3.7.3
Simplify the numerator.
Step 3.7.3.1
Simplify each term.
Step 3.7.3.1.1
Simplify by moving inside the logarithm.
Step 3.7.3.1.2
Simplify each term.
Step 3.7.3.1.2.1
Simplify by moving inside the logarithm.
Step 3.7.3.1.2.2
Raise to the power of .
Step 3.7.3.1.2.3
Simplify by moving inside the logarithm.
Step 3.7.3.1.2.4
Raise to the power of .
Step 3.7.3.1.3
Apply the distributive property.
Step 3.7.3.1.4
Rewrite using the commutative property of multiplication.
Step 3.7.3.1.5
Multiply by by adding the exponents.
Step 3.7.3.1.5.1
Move .
Step 3.7.3.1.5.2
Multiply by .
Step 3.7.3.2
Reorder factors in .
Step 3.7.4
Factor out of .
Step 3.7.5
Factor out of .
Step 3.7.6
Factor out of .
Step 3.7.7
Factor out of .
Step 3.7.8
Factor out of .
Step 3.7.9
Rewrite as .
Step 3.7.10
Factor out of .
Step 3.7.11
Factor out of .
Step 3.7.12
Factor out of .
Step 3.7.13
Rewrite as .
Step 3.7.14
Cancel the common factor.
Step 3.7.15
Rewrite the expression.
Step 3.7.16
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .