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.
Tap for more steps...
Evaluate the function at .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Remove parentheses.
Simplify each term.
Tap for more steps...
Using the Binomial Theorem, expand to .
Apply the distributive property.
Simplify.
Tap for more steps...
Multiply by .
Multiply by .
Remove parentheses.
The final answer is .
Find the components of the definition.
Plug in the components.
Simplify.
Tap for more steps...
Simplify the numerator.
Tap for more steps...
Apply the distributive property.
Simplify.
Tap for more steps...
Multiply by .
Multiply by .
Subtract from .
Add and .
Subtract from .
Add and .
Subtract from .
Add and .
Factor out of .
Tap for more steps...
Factor out of .
Factor out of .
Factor out of .
Raise to the power of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Divide by .
Take the limit of each term.
Tap for more steps...
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 exponent from outside the limit using the Limits Power Rule.
Move the term outside of the limit because it is constant with respect to .
Move the term outside of the limit because it is constant with respect to .
Move the exponent from outside the limit using the Limits Power Rule.
Evaluate the limits by plugging in for all occurrences of .
Tap for more steps...
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 by plugging in for .
Evaluate the limit of which is constant as approaches .
Simplify the answer.
Tap for more steps...
Simplify each term.
Tap for more steps...
Multiply .
Tap for more steps...
Multiply by .
Multiply by .
Raising to any positive power yields .
Multiply by .
Add and .
Add and .
Simplify .
Tap for more steps...
Simplify each term.
Tap for more steps...
One to any power is one.
Multiply by .
Add and .
The slope is and the point is .
Find the value of using the formula for the equation of a line.
Tap for more steps...
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 .
Tap for more steps...
Rewrite the equation as .
Multiply by .
Move all terms not containing to the right side of the equation.
Tap for more steps...
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.
Enter YOUR Problem
Mathway requires javascript and a modern browser.
Cookies & Privacy
This website uses cookies to ensure you get the best experience on our website.
More Information