Calculus Examples

Find the Derivative - d/dx 1/2x^2 square root of 25-x^2
Step 1
Differentiate using the Constant Multiple Rule.
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Step 1.1
Combine and .
Step 1.2
Combine fractions.
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Step 1.2.1
Combine and .
Step 1.2.2
Use to rewrite as .
Step 1.3
Since is constant with respect to , the derivative of with respect to is .
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
Add and .
Step 12
Since is constant with respect to , the derivative of with respect to is .
Step 13
Differentiate using the Power Rule which states that is where .
Step 14
Combine fractions.
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Step 14.1
Multiply by .
Step 14.2
Combine and .
Step 14.3
Combine and .
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
Reorder.
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Step 22.1
Move to the left of .
Step 22.2
Move .
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
Move .
Step 25.2
Use the power rule to combine exponents.
Step 25.3
Combine the numerators over the common denominator.
Step 25.4
Add and .
Step 25.5
Divide by .
Step 26
Simplify .
Step 27
Multiply by .
Step 28
Simplify.
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Step 28.1
Apply the distributive property.
Step 28.2
Simplify the numerator.
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Step 28.2.1
Simplify each term.
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Step 28.2.1.1
Rewrite using the commutative property of multiplication.
Step 28.2.1.2
Multiply by by adding the exponents.
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Step 28.2.1.2.1
Move .
Step 28.2.1.2.2
Multiply by .
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Step 28.2.1.2.2.1
Raise to the power of .
Step 28.2.1.2.2.2
Use the power rule to combine exponents.
Step 28.2.1.2.3
Add and .
Step 28.2.1.3
Multiply by .
Step 28.2.1.4
Multiply by .
Step 28.2.2
Subtract from .
Step 28.3
Factor out of .
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Step 28.3.1
Factor out of .
Step 28.3.2
Factor out of .
Step 28.3.3
Factor out of .
Step 28.4
Factor out of .
Step 28.5
Rewrite as .
Step 28.6
Factor out of .
Step 28.7
Rewrite as .
Step 28.8
Move the negative in front of the fraction.