Calculus Examples

Find Where Increasing/Decreasing Using Derivatives f(x) = square root of x^2+4
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Use to rewrite as .
Step 1.1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.1.2.1
To apply the Chain Rule, set as .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Replace all occurrences of with .
Step 1.1.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.4
Combine and .
Step 1.1.5
Combine the numerators over the common denominator.
Step 1.1.6
Simplify the numerator.
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Step 1.1.6.1
Multiply by .
Step 1.1.6.2
Subtract from .
Step 1.1.7
Combine fractions.
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Step 1.1.7.1
Move the negative in front of the fraction.
Step 1.1.7.2
Combine and .
Step 1.1.7.3
Move to the denominator using the negative exponent rule .
Step 1.1.8
By the Sum Rule, the derivative of with respect to is .
Step 1.1.9
Differentiate using the Power Rule which states that is where .
Step 1.1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.11
Simplify terms.
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Step 1.1.11.1
Add and .
Step 1.1.11.2
Combine and .
Step 1.1.11.3
Combine and .
Step 1.1.11.4
Cancel the common factor.
Step 1.1.11.5
Rewrite the expression.
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
Set the numerator equal to zero.
Step 3
The values which make the derivative equal to are .
Step 4
After finding the point that makes the derivative equal to or undefined, the interval to check where is increasing and where it is decreasing is .
Step 5
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
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Step 5.2.1
Simplify the denominator.
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Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Add and .
Step 5.2.2
Move the negative in front of the fraction.
Step 5.2.3
The final answer is .
Step 5.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 6
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
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Step 6.2.1
Simplify the denominator.
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Step 6.2.1.1
One to any power is one.
Step 6.2.1.2
Add and .
Step 6.2.2
The final answer is .
Step 6.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 7
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 8