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.
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Evaluate the function at .
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Replace the variable with in the expression.
Simplify the result.
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Remove parentheses.
Remove unnecessary parentheses.
Simplify each term.
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Using the Binomial Theorem, expand to .
Apply the distributive property.
Simplify.
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Multiply by to get .
Multiply by to get .
Simplify each term.
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Remove parentheses around .
Remove parentheses around .
The final answer is .
Find the components of the definition.
Plug in the components.
Simplify.
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Simplify the numerator.
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Apply the distributive property.
Simplify.
Move parentheses.
Remove parentheses.
Subtract from to get .
Add and to get .
Subtract from to get .
Add and to get .
Subtract from to get .
Add and to get .
Factor out of .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by to get .
Take the limit of each term.
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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 .
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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.
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Simplify each term.
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Remove parentheses around .
Multiply by to get .
Simplify .
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Multiply by to get .
Multiply by to get .
Remove parentheses around .
Raising to any positive power yields .
Multiply by to get .
Add and to get .
Add and to get .
Find the slope . In this case .
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Simplify each term.
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Remove parentheses around .
One to any power is one.
Multiply by to get .
Add and to get .
The slope is and the point is .
Find the value of using the formula for the equation of a line.
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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 .
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Rewrite the equation as .
Simplify each term.
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Multiply by to get .
Multiply by to get .
Move all terms not containing to the right side of the equation.
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Since does not contain the variable to solve for, move it to the right side of the equation by subtracting from both sides.
Add and to get .
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.
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