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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
Step 2.1
Differentiate using the chain rule, which states that is where and .
Step 2.1.1
To apply the Chain Rule, set as .
Step 2.1.2
The derivative of with respect to is .
Step 2.1.3
Replace all occurrences of with .
Step 2.2
Differentiate using the Product Rule which states that is where and .
Step 2.3
Differentiate using the chain rule, which states that is where and .
Step 2.3.1
To apply the Chain Rule, set as .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Move to the left of .
Step 2.5
Rewrite as .
Step 2.6
Differentiate using the Power Rule which states that is where .
Step 2.7
Multiply by .
Step 2.8
Simplify.
Step 2.8.1
Apply the distributive property.
Step 2.8.2
Reorder terms.
Step 3
Step 3.1
Differentiate using the chain rule, which states that is where and .
Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate.
Step 3.2.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Rewrite as .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Step 5.1
Simplify the left side.
Step 5.1.1
Reorder factors in .
Step 5.2
Simplify the right side.
Step 5.2.1
Simplify .
Step 5.2.1.1
Apply the distributive property.
Step 5.2.1.2
Simplify the expression.
Step 5.2.1.2.1
Rewrite using the commutative property of multiplication.
Step 5.2.1.2.2
Reorder factors in .
Step 5.3
Subtract from both sides of the equation.
Step 5.4
Subtract from both sides of the equation.
Step 5.5
Factor out of .
Step 5.5.1
Factor out of .
Step 5.5.2
Factor out of .
Step 5.5.3
Factor out of .
Step 5.6
Rewrite as .
Step 5.7
Divide each term in by and simplify.
Step 5.7.1
Divide each term in by .
Step 5.7.2
Simplify the left side.
Step 5.7.2.1
Cancel the common factor of .
Step 5.7.2.1.1
Cancel the common factor.
Step 5.7.2.1.2
Divide by .
Step 5.7.3
Simplify the right side.
Step 5.7.3.1
Move the negative in front of the fraction.
Step 5.7.3.2
Combine the numerators over the common denominator.
Step 6
Replace with .