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Calculus Examples
Step 1
Consider the limit definition of the derivative.
Step 2
Step 2.1
Evaluate the function at .
Step 2.1.1
Replace the variable with in the expression.
Step 2.1.2
Simplify the result.
Step 2.1.2.1
Use the Binomial Theorem.
Step 2.1.2.2
The final answer is .
Step 2.2
Reorder.
Step 2.2.1
Move .
Step 2.2.2
Move .
Step 2.2.3
Move .
Step 2.2.4
Move .
Step 2.2.5
Reorder and .
Step 2.3
Find the components of the definition.
Step 3
Plug in the components.
Step 4
Step 4.1
Simplify the numerator.
Step 4.1.1
Subtract from .
Step 4.1.2
Add and .
Step 4.1.3
Factor out of .
Step 4.1.3.1
Factor out of .
Step 4.1.3.2
Factor out of .
Step 4.1.3.3
Factor out of .
Step 4.1.3.4
Factor out of .
Step 4.1.3.5
Factor out of .
Step 4.2
Reduce the expression by cancelling the common factors.
Step 4.2.1
Cancel the common factor of .
Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.2.2
Simplify the expression.
Step 4.2.2.1
Move .
Step 4.2.2.2
Move .
Step 4.2.2.3
Reorder and .
Step 5
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 6
Evaluate the limit of which is constant as approaches .
Step 7
Move the term outside of the limit because it is constant with respect to .
Step 8
Move the exponent from outside the limit using the Limits Power Rule.
Step 9
Step 9.1
Evaluate the limit of by plugging in for .
Step 9.2
Evaluate the limit of by plugging in for .
Step 10
Step 10.1
Simplify each term.
Step 10.1.1
Multiply .
Step 10.1.1.1
Multiply by .
Step 10.1.1.2
Multiply by .
Step 10.1.2
Raising to any positive power yields .
Step 10.2
Combine the opposite terms in .
Step 10.2.1
Add and .
Step 10.2.2
Add and .
Step 11