Calculus Examples

Find the Derivative - d/dx x* square root of 8x-x^2
Step 1
Use to rewrite as .
Step 2
Differentiate using the Product Rule which states that is where and .
Step 3
Differentiate using the chain rule, which states that is where and .
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Step 3.1
To apply the Chain Rule, set as .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Replace all occurrences of with .
Step 4
To write as a fraction with a common denominator, multiply by .
Step 5
Combine and .
Step 6
Combine the numerators over the common denominator.
Step 7
Simplify the numerator.
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Step 7.1
Multiply by .
Step 7.2
Subtract from .
Step 8
Combine fractions.
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Step 8.1
Move the negative in front of the fraction.
Step 8.2
Combine and .
Step 8.3
Move to the denominator using the negative exponent rule .
Step 8.4
Combine and .
Step 9
By the Sum Rule, the derivative of with respect to is .
Step 10
Since is constant with respect to , the derivative of with respect to is .
Step 11
Differentiate using the Power Rule which states that is where .
Step 12
Multiply by .
Step 13
Since is constant with respect to , the derivative of with respect to is .
Step 14
Differentiate using the Power Rule which states that is where .
Step 15
Multiply by .
Step 16
Differentiate using the Power Rule which states that is where .
Step 17
Multiply by .
Step 18
Simplify.
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Step 18.1
Reorder terms.
Step 18.2
Simplify each term.
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Step 18.2.1
Multiply by .
Step 18.2.2
Factor out of .
Step 18.2.3
Cancel the common factors.
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Step 18.2.3.1
Factor out of .
Step 18.2.3.2
Cancel the common factor.
Step 18.2.3.3
Rewrite the expression.
Step 18.3
To write as a fraction with a common denominator, multiply by .
Step 18.4
Combine the numerators over the common denominator.
Step 18.5
Simplify the numerator.
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Step 18.5.1
Apply the distributive property.
Step 18.5.2
Multiply by by adding the exponents.
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Step 18.5.2.1
Move .
Step 18.5.2.2
Multiply by .
Step 18.5.3
Multiply by by adding the exponents.
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Step 18.5.3.1
Use the power rule to combine exponents.
Step 18.5.3.2
Combine the numerators over the common denominator.
Step 18.5.3.3
Add and .
Step 18.5.3.4
Divide by .
Step 18.5.4
Simplify .
Step 18.5.5
Add and .
Step 18.5.6
Subtract from .
Step 18.5.7
Factor out of .
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Step 18.5.7.1
Factor out of .
Step 18.5.7.2
Factor out of .
Step 18.5.7.3
Factor out of .
Step 18.6
Factor out of .
Step 18.7
Rewrite as .
Step 18.8
Factor out of .
Step 18.9
Rewrite as .
Step 18.10
Move the negative in front of the fraction.