Calculus Examples

Use the Limit Definition to Find the Derivative
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.
Using the Binomial Theorem, expand to .
Find the components of the definition.
Plug in the components.
Simplify.
Simplify the numerator.
Apply the distributive property.
Multiply by to get .
Remove parentheses.
Subtract from to get .
Factor out of .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide 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 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.
Move the exponent from outside the limit using the Limits Power Rule.
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 by plugging in for .
Evaluate the limit of by plugging in for .
Simplify each term.
Remove parentheses around .
Multiply by to get .
Simplify .
Multiply by to get .
Multiply by to get .
Remove parentheses around .
Raising to any positive power yields .
Simplify .
Multiply by to get .
Multiply by to get .
Remove parentheses around .
Raising to any positive power yields .

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