Calculus Examples

Find the Derivative - d/dx x(16-x^2)^(1/2)
Step 1
Differentiate using the Product Rule which states that is where and .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Replace all occurrences of with .
Step 3
To write as a fraction with a common denominator, multiply by .
Step 4
Combine and .
Step 5
Combine the numerators over the common denominator.
Step 6
Simplify the numerator.
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Step 6.1
Multiply by .
Step 6.2
Subtract from .
Step 7
Combine fractions.
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Step 7.1
Move the negative in front of the fraction.
Step 7.2
Combine and .
Step 7.3
Move to the denominator using the negative exponent rule .
Step 7.4
Combine and .
Step 8
By the Sum Rule, the derivative of with respect to is .
Step 9
Since is constant with respect to , the derivative of with respect to is .
Step 10
Add and .
Step 11
Since is constant with respect to , the derivative of with respect to is .
Step 12
Differentiate using the Power Rule which states that is where .
Step 13
Combine fractions.
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Step 13.1
Multiply by .
Step 13.2
Combine and .
Step 13.3
Combine and .
Step 14
Raise to the power of .
Step 15
Raise to the power of .
Step 16
Use the power rule to combine exponents.
Step 17
Add and .
Step 18
Factor out of .
Step 19
Cancel the common factors.
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Step 19.1
Factor out of .
Step 19.2
Cancel the common factor.
Step 19.3
Rewrite the expression.
Step 20
Move the negative in front of the fraction.
Step 21
Differentiate using the Power Rule which states that is where .
Step 22
Multiply by .
Step 23
To write as a fraction with a common denominator, multiply by .
Step 24
Combine the numerators over the common denominator.
Step 25
Multiply by by adding the exponents.
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Step 25.1
Use the power rule to combine exponents.
Step 25.2
Combine the numerators over the common denominator.
Step 25.3
Add and .
Step 25.4
Divide by .
Step 26
Simplify .
Step 27
Subtract from .
Step 28
Reorder terms.