# 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.
Apply the distributive property.
Find the components of the definition.
Plug in the components.
Simplify.
Simplify the numerator.
Group the last three terms.
Factor.
Multiply by to get .
Remove parentheses.
Remove parentheses.
Factor out of .
Apply the distributive property.
Multiply by to get .
Multiply by to get .
Remove parentheses.
Subtract from to get .
Subtract from to get .
Factor.
Multiply by to get .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by to get .
Evaluate the limit of which is constant as approaches .