Calculus Examples

Use the Limit Definition to Find the Derivative 1/x
Step 1
Consider the limit definition of the derivative.
Step 2
Find the components of the definition.
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Step 2.1
Evaluate the function at .
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Step 2.1.1
Replace the variable with in the expression.
Step 2.1.2
The final answer is .
Step 2.2
Find the components of the definition.
Step 3
Plug in the components.
Step 4
Simplify.
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Step 4.1
Simplify the numerator.
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Step 4.1.1
To write as a fraction with a common denominator, multiply by .
Step 4.1.2
To write as a fraction with a common denominator, multiply by .
Step 4.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.1.3.1
Multiply by .
Step 4.1.3.2
Multiply by .
Step 4.1.3.3
Reorder the factors of .
Step 4.1.4
Combine the numerators over the common denominator.
Step 4.1.5
Rewrite in a factored form.
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Step 4.1.5.1
Apply the distributive property.
Step 4.1.5.2
Subtract from .
Step 4.1.5.3
Subtract from .
Step 4.1.6
Move the negative in front of the fraction.
Step 4.2
Multiply the numerator by the reciprocal of the denominator.
Step 4.3
Cancel the common factor of .
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Step 4.3.1
Move the leading negative in into the numerator.
Step 4.3.2
Factor out of .
Step 4.3.3
Cancel the common factor.
Step 4.3.4
Rewrite the expression.
Step 4.4
Move the negative in front of the fraction.
Step 5
Evaluate the limit.
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Step 5.1
Move the term outside of the limit because it is constant with respect to .
Step 5.2
Move the term outside of the limit because it is constant with respect to .
Step 5.3
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 5.4
Evaluate the limit of which is constant as approaches .
Step 5.5
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 5.6
Evaluate the limit of which is constant as approaches .
Step 6
Evaluate the limit of by plugging in for .
Step 7
Simplify the answer.
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Step 7.1
Add and .
Step 7.2
Multiply .
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Step 7.2.1
Multiply by .
Step 7.2.2
Raise to the power of .
Step 7.2.3
Raise to the power of .
Step 7.2.4
Use the power rule to combine exponents.
Step 7.2.5
Add and .
Step 8