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Calculus Examples
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
By the Sum Rule, the derivative of with respect to is .
Step 3
Step 3.1
To apply the Chain Rule, set as .
Step 3.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.3
Replace all occurrences of with .
Step 4
Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
Simplify the expression.
Step 4.3.1
Multiply by .
Step 4.3.2
Move to the left of .
Step 4.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.5
Simplify the expression.
Step 4.5.1
Add and .
Step 4.5.2
Move to the left of .
Step 4.6
By the Sum Rule, the derivative of with respect to is .
Step 5
Step 5.1
To apply the Chain Rule, set as .
Step 5.2
Differentiate using the Exponential Rule which states that is where =.
Step 5.3
Replace all occurrences of with .
Step 6
Step 6.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Simplify the expression.
Step 6.3.1
Multiply by .
Step 6.3.2
Move to the left of .
Step 6.4
Since is constant with respect to , the derivative of with respect to is .
Step 6.5
Simplify the expression.
Step 6.5.1
Add and .
Step 6.5.2
Multiply by .
Step 7
Step 7.1
Apply the distributive property.
Step 7.2
Apply the distributive property.
Step 7.3
Apply the distributive property.
Step 7.4
Apply the distributive property.
Step 7.5
Simplify the numerator.
Step 7.5.1
Combine the opposite terms in .
Step 7.5.1.1
Subtract from .
Step 7.5.1.2
Add and .
Step 7.5.2
Simplify each term.
Step 7.5.2.1
Multiply by .
Step 7.5.2.2
Multiply by .
Step 7.5.3
Subtract from .
Step 7.6
Move the negative in front of the fraction.
Step 7.7
Simplify the denominator.
Step 7.7.1
Rewrite as .
Step 7.7.2
Rewrite as .
Step 7.7.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 7.7.4
Apply the product rule to .