# Calculus Examples

Use the Limit Definition to Find the Derivative
Consider the limit definition of the derivative.
Find the components of the definition.
Evaluate the function at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Reorder and .
Apply the distributive property.
Reorder.
Move .
Move .
Reorder and .
Find the components of the definition.
Plug in the components.
Simplify.
Simplify the numerator.
Apply the distributive property.
Multiply by .
Subtract from .
Subtract from .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Reduce the expression by cancelling the common factors.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Reorder and .
Take the limit of each term.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Split the limit using the Product of Limits Rule on the limit as approaches .
Evaluate the limits by plugging in for all occurrences of .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .