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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
Differentiate using the Exponential Rule which states that is where =.
Step 2.1.3
Replace all occurrences of with .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Differentiate using the Power Rule.
Step 2.3.1
Differentiate using the Power Rule which states that is where .
Step 2.3.2
Multiply by .
Step 2.4
Rewrite as .
Step 2.5
Combine and .
Step 3
Step 3.1
Differentiate.
Step 3.1.1
By the Sum Rule, the derivative of with respect to is .
Step 3.1.2
Differentiate using the Power Rule which states that is where .
Step 3.2
Evaluate .
Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Rewrite as .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Step 5.1
Multiply both sides by .
Step 5.2
Simplify.
Step 5.2.1
Simplify the left side.
Step 5.2.1.1
Simplify .
Step 5.2.1.1.1
Cancel the common factor of .
Step 5.2.1.1.1.1
Cancel the common factor.
Step 5.2.1.1.1.2
Rewrite the expression.
Step 5.2.1.1.2
Apply the distributive property.
Step 5.2.1.1.3
Simplify the expression.
Step 5.2.1.1.3.1
Rewrite using the commutative property of multiplication.
Step 5.2.1.1.3.2
Reorder factors in .
Step 5.2.1.1.3.3
Move .
Step 5.2.1.1.3.4
Reorder and .
Step 5.2.2
Simplify the right side.
Step 5.2.2.1
Simplify .
Step 5.2.2.1.1
Apply the distributive property.
Step 5.2.2.1.2
Simplify the expression.
Step 5.2.2.1.2.1
Multiply by .
Step 5.2.2.1.2.2
Reorder and .
Step 5.3
Solve for .
Step 5.3.1
Add to both sides of the equation.
Step 5.3.2
Subtract from both sides of the equation.
Step 5.3.3
Factor out of .
Step 5.3.3.1
Factor out of .
Step 5.3.3.2
Factor out of .
Step 5.3.3.3
Factor out of .
Step 5.3.4
Rewrite as .
Step 5.3.5
Divide each term in by and simplify.
Step 5.3.5.1
Divide each term in by .
Step 5.3.5.2
Simplify the left side.
Step 5.3.5.2.1
Cancel the common factor of .
Step 5.3.5.2.1.1
Cancel the common factor.
Step 5.3.5.2.1.2
Divide by .
Step 5.3.5.3
Simplify the right side.
Step 5.3.5.3.1
Combine the numerators over the common denominator.
Step 5.3.5.3.2
Simplify the numerator.
Step 5.3.5.3.2.1
Factor out of .
Step 5.3.5.3.2.1.1
Factor out of .
Step 5.3.5.3.2.1.2
Factor out of .
Step 5.3.5.3.2.1.3
Factor out of .
Step 5.3.5.3.2.2
Rewrite as .
Step 5.3.5.3.3
Simplify with factoring out.
Step 5.3.5.3.3.1
Factor out of .
Step 5.3.5.3.3.2
Factor out of .
Step 5.3.5.3.3.3
Factor out of .
Step 5.3.5.3.3.4
Rewrite negatives.
Step 5.3.5.3.3.4.1
Rewrite as .
Step 5.3.5.3.3.4.2
Move the negative in front of the fraction.
Step 6
Replace with .