# Calculus Examples

Find the Tangent at a Given Point Using the Limit Definition
,
The slope of the tangent line is the derivative of the expression.
The derivative of
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.
Remove parentheses.
Simplify and combine like terms.
Simplify each term.
Simplify .
Use the power rule to combine exponents.
Simplify .
Use the power rule to combine exponents.
Reorder and .
Apply the distributive property.
Multiply by .
Remove parentheses around .
Apply the distributive property.
Remove unnecessary parentheses.
Find the components of the definition.
Plug in the components.
Simplify.
Simplify the numerator.
Apply the distributive property.
Multiply by .
Multiply by .
Remove parentheses.
Subtract from .
Subtract from .
Factor out of .
Simplify terms.
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by .
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
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 .
Move the term outside of the limit because it is constant with respect to .
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 .
Simplify each term.
Multiply by .
Multiply by .
Find the slope . In this case .
Multiply by .
The slope is and the point is .
Find the value of using the formula for the equation of a line.
Use the formula for the equation of a line to find .
Substitute the value of into the equation.
Substitute the value of into the equation.
Substitute the value of into the equation.
Find the value of .
Rewrite the equation as .
Simplify each term.
Multiply by .
Multiply by .
Move all terms not containing to the right side of the equation.
Subtract from both sides of the equation.