Calculus Examples

Check if Differentiable Over an Interval
,
Find the derivative.
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Find the first derivative.
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Rewrite as .
Differentiate using the Power Rule which states that is where .
Remove the negative exponent by rewriting as .
The derivative of with respect to is .
Find if the derivative is continuous on .
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To find whether the function is continuous on or not, find the domain of .
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Solve to find the value of that makes the expression undefined.
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Set up the equation to solve for .
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
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Simplify the right side of the equation.
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Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
is equal to .
For asymptotes and point discontinuity (denominator equals ), it is easier to find where the expression is undefined. These values are not part of the domain.
The domain is all values of that make the expression defined.
is not continuous on because is not in the domain of .
The function is not continuous.
The function is not continuous.
The function is not differentiable on because the derivative is not continuous on .
The function is not differentiable.
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