# 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.
Use the power rule to combine exponents.
Add and to get .
Use the power rule to combine exponents.
Add and to get .
Add and to get .
Reorder and .
Add and to get .
Apply the distributive property.
Multiply by to get .
Remove parentheses around .
Apply the distributive property.
Remove unnecessary parentheses.
The final answer is .
Find the components of the definition.
Plug in the components.
Simplify.
Simplify the numerator.
Apply the distributive property.
Multiply by to get .
Multiply by to get .
Remove parentheses.
Subtract from to get .
Add and to get .
Subtract from to get .
Add and to get .
Factor out of .
Simplify terms.
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by to get .
Apply the distributive property.
Simplify.
Multiply by to get .
Multiply by to get .
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 to get .
Multiply by to get .
Add and to get .
Find the slope . In this case .
Multiply by to get .
Add and to get .
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 to get .
Multiply by to get .
Move all terms not containing to the right side of the equation.
Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Add and to get .
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.