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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
Use the properties of logarithms to simplify the differentiation.
Step 3.1.1
Rewrite as .
Step 3.1.2
Expand by moving outside the logarithm.
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 Exponential Rule which states that is where =.
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Differentiate using the Product Rule which states that is where and .
Step 3.5
Differentiate using the chain rule, which states that is where and .
Step 3.5.1
To apply the Chain Rule, set as .
Step 3.5.2
The derivative of with respect to is .
Step 3.5.3
Replace all occurrences of with .
Step 3.6
Differentiate.
Step 3.6.1
Combine and .
Step 3.6.2
By the Sum Rule, the derivative of with respect to is .
Step 3.6.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.4
Differentiate using the Power Rule which states that is where .
Step 3.6.5
Multiply by .
Step 3.6.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.7
Combine fractions.
Step 3.6.7.1
Add and .
Step 3.6.7.2
Combine and .
Step 3.6.7.3
Move to the left of .
Step 3.6.8
Differentiate using the Power Rule which states that is where .
Step 3.6.9
Multiply by .
Step 3.7
To write as a fraction with a common denominator, multiply by .
Step 3.8
Combine the numerators over the common denominator.
Step 3.9
Combine and .
Step 3.10
Combine and .
Step 3.11
Simplify.
Step 3.11.1
Apply the distributive property.
Step 3.11.2
Simplify the numerator.
Step 3.11.2.1
Simplify by moving inside the logarithm.
Step 3.11.2.2
Simplify each term.
Step 3.11.2.2.1
Multiply by .
Step 3.11.2.2.2
Apply the distributive property.
Step 3.11.2.2.3
Rewrite using the commutative property of multiplication.
Step 3.11.2.2.4
Multiply by .
Step 3.11.2.2.5
Simplify by moving inside the logarithm.
Step 3.11.2.2.6
Apply the distributive property.
Step 3.11.2.2.7
Multiply .
Step 3.11.2.2.7.1
Reorder and .
Step 3.11.2.2.7.2
Simplify by moving inside the logarithm.
Step 3.11.2.2.8
Simplify by moving inside the logarithm.
Step 3.11.2.2.9
Multiply the exponents in .
Step 3.11.2.2.9.1
Apply the power rule and multiply exponents, .
Step 3.11.2.2.9.2
Multiply by .
Step 3.11.2.3
Apply the distributive property.
Step 3.11.2.4
Rewrite using the commutative property of multiplication.
Step 3.11.2.5
Reorder factors in .
Step 3.11.3
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .