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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Rewrite as .
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify each term.
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Multiply by .
Move to the left of .
Multiply by .
Move to the left of .
Multiply by .
Add and .
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Reorder and .
Add and .
Add and .
Add and .
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.
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Multiply by .
Multiply by .
Subtract from .
Add and .
Subtract from .
Add and .
Subtract from .
Add and .
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Divide by .
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 .
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 which is constant as approaches .
Add and .
Simplify .
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Multiply by .
Add and .
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 .
Multiply by .
Move all terms not containing to the right side of the equation.
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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.
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