# Calculus Examples

Use the Limit Definition to Find the Derivative
Step 1
Consider the limit definition of the derivative.
Step 2
Find the components of the definition.
Step 2.1
Evaluate the function at .
Step 2.1.1
Replace the variable with in the expression.
Step 2.1.2
Simplify the result.
Step 2.1.2.1
Simplify each term.
Step 2.1.2.1.1
Rewrite as .
Step 2.1.2.1.2
Expand using the FOIL Method.
Step 2.1.2.1.2.1
Apply the distributive property.
Step 2.1.2.1.2.2
Apply the distributive property.
Step 2.1.2.1.2.3
Apply the distributive property.
Step 2.1.2.1.3
Simplify and combine like terms.
Step 2.1.2.1.3.1
Simplify each term.
Step 2.1.2.1.3.1.1
Multiply by .
Step 2.1.2.1.3.1.2
Multiply by .
Step 2.1.2.1.3.2
Step 2.1.2.1.3.2.1
Reorder and .
Step 2.1.2.1.3.2.2
Step 2.1.2.2
Step 2.2
Reorder.
Step 2.2.1
Move .
Step 2.2.2
Reorder and .
Step 2.3
Find the components of the definition.
Step 3
Plug in the components.
Step 4
Simplify.
Step 4.1
Simplify the numerator.
Step 4.1.1
Apply the distributive property.
Step 4.1.2
Multiply by .
Step 4.1.3
Subtract from .
Step 4.1.4
Step 4.1.5
Step 4.1.6
Step 4.1.7
Factor out of .
Step 4.1.7.1
Factor out of .
Step 4.1.7.2
Factor out of .
Step 4.1.7.3
Factor out of .
Step 4.2
Reduce the expression by cancelling the common factors.
Step 4.2.1
Cancel the common factor of .
Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.2.2
Reorder and .
Step 5
Evaluate the limit.
Step 5.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 5.2
Evaluate the limit of which is constant as approaches .
Step 6
Evaluate the limit of by plugging in for .
Step 7