# 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.
Rewrite as .
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify each term.
Multiply by .
Move to the left of .
Multiply by .
Move to the left of .
Multiply by .
Reorder and .
Find the components of the definition.
Plug in the components.
Simplify.
Simplify the numerator.
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
Subtract from .
Subtract from .
Subtract from .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Divide 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 .
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 .
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 .
Multiply by .
Move all terms not containing to the right side of the equation.
Subtract from both sides of the equation.
Subtract from .
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.