Calculus Examples
,
The slope of the tangent line is the derivative of the expression.
The derivative of
Consider the limit definition of the derivative.
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 .
Add and .
Reorder and .
Add and .
Apply the distributive property.
The final answer is .
Find the components of the definition.
Plug in the components.
Simplify the numerator.
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
Subtract from .
Add and .
Subtract from .
Add and .
Subtract from .
Add and .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Divide by .
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 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 .
Add and .
Multiply by .
Add and .
The slope is and the point is .
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 the left side.
Multiply by .
Add and .
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.