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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
Differentiate using the Product Rule which states that is where and .
Step 2.2.2
Differentiate using the chain rule, which states that is where and .
Step 2.2.2.1
To apply the Chain Rule, set as .
Step 2.2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.2.3
Replace all occurrences of with .
Step 2.2.3
Rewrite as .
Step 2.2.4
Use the properties of logarithms to simplify the differentiation.
Step 2.2.4.1
Rewrite as .
Step 2.2.4.2
Expand by moving outside the logarithm.
Step 2.2.5
Differentiate using the chain rule, which states that is where and .
Step 2.2.5.1
To apply the Chain Rule, set as .
Step 2.2.5.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.2.5.3
Replace all occurrences of with .
Step 2.2.6
Differentiate using the Product Rule which states that is where and .
Step 2.2.7
The derivative of with respect to is .
Step 2.2.8
Differentiate using the Power Rule which states that is where .
Step 2.2.9
Move to the left of .
Step 2.2.10
Combine and .
Step 2.2.11
Cancel the common factor of .
Step 2.2.11.1
Cancel the common factor.
Step 2.2.11.2
Rewrite the expression.
Step 2.2.12
Multiply by .
Step 2.3
Evaluate .
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Rewrite as .
Step 2.4
Simplify.
Step 2.4.1
Apply the distributive property.
Step 2.4.2
Multiply by .
Step 2.4.3
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 Power Rule which states that is where .
Step 3.3
Multiply by .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Step 5.1
Move all the terms containing a logarithm to the left side of the equation.
Step 5.2
Reorder factors in .
Step 5.3
Reorder factors in .
Step 5.4
Rewrite the equation as .
Step 5.5
Reorder factors in .
Step 5.6
Subtract from both sides of the equation.
Step 5.7
Factor out of .
Step 5.7.1
Factor out of .
Step 5.7.2
Factor out of .
Step 5.7.3
Factor out of .
Step 5.8
Divide each term in by and simplify.
Step 5.8.1
Divide each term in by .
Step 5.8.2
Simplify the left side.
Step 5.8.2.1
Cancel the common factor of .
Step 5.8.2.1.1
Cancel the common factor.
Step 5.8.2.1.2
Divide by .
Step 5.8.3
Simplify the right side.
Step 5.8.3.1
Move the negative in front of the fraction.
Step 5.8.3.2
Combine the numerators over the common denominator.
Step 5.8.3.3
Combine the numerators over the common denominator.
Step 5.8.3.4
Factor out of .
Step 5.8.3.5
Rewrite as .
Step 5.8.3.6
Factor out of .
Step 5.8.3.7
Rewrite negatives.
Step 5.8.3.7.1
Rewrite as .
Step 5.8.3.7.2
Move the negative in front of the fraction.
Step 6
Replace with .