Calculus Examples

Find the Derivative - d/dx y=x/( square root of 4-x^2)-arcsin(x/2)
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Evaluate .
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Step 2.1
Use to rewrite as .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Differentiate using the chain rule, which states that is where and .
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Step 2.4.1
To apply the Chain Rule, set as .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Replace all occurrences of with .
Step 2.5
By the Sum Rule, the derivative of with respect to is .
Step 2.6
Since is constant with respect to , the derivative of with respect to is .
Step 2.7
Since is constant with respect to , the derivative of with respect to is .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Multiply by .
Step 2.10
To write as a fraction with a common denominator, multiply by .
Step 2.11
Combine and .
Step 2.12
Combine the numerators over the common denominator.
Step 2.13
Simplify the numerator.
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Step 2.13.1
Multiply by .
Step 2.13.2
Subtract from .
Step 2.14
Move the negative in front of the fraction.
Step 2.15
Multiply by .
Step 2.16
Subtract from .
Step 2.17
Combine and .
Step 2.18
Combine and .
Step 2.19
Combine and .
Step 2.20
Move to the denominator using the negative exponent rule .
Step 2.21
Factor out of .
Step 2.22
Cancel the common factors.
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Step 2.22.1
Factor out of .
Step 2.22.2
Cancel the common factor.
Step 2.22.3
Rewrite the expression.
Step 2.23
Move the negative in front of the fraction.
Step 2.24
Multiply by .
Step 2.25
Multiply by .
Step 2.26
Combine and .
Step 2.27
Raise to the power of .
Step 2.28
Raise to the power of .
Step 2.29
Use the power rule to combine exponents.
Step 2.30
Add and .
Step 2.31
To write as a fraction with a common denominator, multiply by .
Step 2.32
Combine the numerators over the common denominator.
Step 2.33
Multiply by by adding the exponents.
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Step 2.33.1
Use the power rule to combine exponents.
Step 2.33.2
Combine the numerators over the common denominator.
Step 2.33.3
Add and .
Step 2.33.4
Divide by .
Step 2.34
Simplify .
Step 2.35
Add and .
Step 2.36
Add and .
Step 2.37
Multiply the exponents in .
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Step 2.37.1
Apply the power rule and multiply exponents, .
Step 2.37.2
Cancel the common factor of .
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Step 2.37.2.1
Cancel the common factor.
Step 2.37.2.2
Rewrite the expression.
Step 2.38
Simplify.
Step 2.39
Rewrite as a product.
Step 2.40
Multiply by .
Step 2.41
Multiply by by adding the exponents.
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Step 2.41.1
Multiply by .
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Step 2.41.1.1
Raise to the power of .
Step 2.41.1.2
Use the power rule to combine exponents.
Step 2.41.2
Write as a fraction with a common denominator.
Step 2.41.3
Combine the numerators over the common denominator.
Step 2.41.4
Add and .
Step 3
Evaluate .
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Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Multiply by .
Step 3.6
Multiply by .
Step 3.7
Move to the left of .
Step 4
Simplify.
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Step 4.1
Apply the product rule to .
Step 4.2
Raise to the power of .
Step 4.3
Simplify each term.
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Step 4.3.1
Simplify the denominator.
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Step 4.3.1.1
Write as a fraction with a common denominator.
Step 4.3.1.2
Combine the numerators over the common denominator.
Step 4.3.1.3
Rewrite in a factored form.
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Step 4.3.1.3.1
Rewrite as .
Step 4.3.1.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.1.4
Rewrite as .
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Step 4.3.1.4.1
Factor the perfect power out of .
Step 4.3.1.4.2
Factor the perfect power out of .
Step 4.3.1.4.3
Rearrange the fraction .
Step 4.3.1.5
Pull terms out from under the radical.
Step 4.3.1.6
Combine and .
Step 4.3.2
Combine and .
Step 4.3.3
Reduce the expression by cancelling the common factors.
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Step 4.3.3.1
Reduce the expression by cancelling the common factors.
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Step 4.3.3.1.1
Cancel the common factor.
Step 4.3.3.1.2
Rewrite the expression.
Step 4.3.3.2
Divide by .
Step 4.3.4
Multiply by .
Step 4.3.5
Combine and simplify the denominator.
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Step 4.3.5.1
Multiply by .
Step 4.3.5.2
Raise to the power of .
Step 4.3.5.3
Raise to the power of .
Step 4.3.5.4
Use the power rule to combine exponents.
Step 4.3.5.5
Add and .
Step 4.3.5.6
Rewrite as .
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Step 4.3.5.6.1
Use to rewrite as .
Step 4.3.5.6.2
Apply the power rule and multiply exponents, .
Step 4.3.5.6.3
Combine and .
Step 4.3.5.6.4
Cancel the common factor of .
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Step 4.3.5.6.4.1
Cancel the common factor.
Step 4.3.5.6.4.2
Rewrite the expression.
Step 4.3.5.6.5
Simplify.
Step 4.4
To write as a fraction with a common denominator, multiply by .
Step 4.5
To write as a fraction with a common denominator, multiply by .
Step 4.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.6.1
Multiply by .
Step 4.6.2
Multiply by .
Step 4.6.3
Reorder the factors of .
Step 4.6.4
Reorder the factors of .
Step 4.7
Combine the numerators over the common denominator.
Step 4.8
Simplify the numerator.
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Step 4.8.1
Use to rewrite as .
Step 4.8.2
Apply the distributive property.
Step 4.8.3
Multiply by .
Step 4.8.4
Expand using the FOIL Method.
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Step 4.8.4.1
Apply the distributive property.
Step 4.8.4.2
Apply the distributive property.
Step 4.8.4.3
Apply the distributive property.
Step 4.8.5
Simplify and combine like terms.
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Step 4.8.5.1
Simplify each term.
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Step 4.8.5.1.1
Multiply by .
Step 4.8.5.1.2
Multiply by .
Step 4.8.5.1.3
Multiply by .
Step 4.8.5.1.4
Rewrite using the commutative property of multiplication.
Step 4.8.5.1.5
Multiply by by adding the exponents.
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Step 4.8.5.1.5.1
Move .
Step 4.8.5.1.5.2
Multiply by .
Step 4.8.5.1.6
Multiply by .
Step 4.8.5.2
Add and .
Step 4.8.5.3
Add and .
Step 4.8.6
Expand using the FOIL Method.
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Step 4.8.6.1
Apply the distributive property.
Step 4.8.6.2
Apply the distributive property.
Step 4.8.6.3
Apply the distributive property.
Step 4.8.7
Simplify and combine like terms.
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Step 4.8.7.1
Simplify each term.
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Step 4.8.7.1.1
Multiply by .
Step 4.8.7.1.2
Multiply by .
Step 4.8.7.1.3
Move to the left of .
Step 4.8.7.1.4
Rewrite using the commutative property of multiplication.
Step 4.8.7.1.5
Multiply by by adding the exponents.
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Step 4.8.7.1.5.1
Move .
Step 4.8.7.1.5.2
Multiply by .
Step 4.8.7.2
Add and .
Step 4.8.7.3
Add and .
Step 4.8.8
Multiply by by adding the exponents.
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Step 4.8.8.1
Move .
Step 4.8.8.2
Use the power rule to combine exponents.
Step 4.8.8.3
Combine the numerators over the common denominator.
Step 4.8.8.4
Add and .
Step 4.8.8.5
Divide by .
Step 4.8.9
Rewrite as .
Step 4.8.10
Expand using the FOIL Method.
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Step 4.8.10.1
Apply the distributive property.
Step 4.8.10.2
Apply the distributive property.
Step 4.8.10.3
Apply the distributive property.
Step 4.8.11
Simplify and combine like terms.
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Step 4.8.11.1
Simplify each term.
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Step 4.8.11.1.1
Multiply by .
Step 4.8.11.1.2
Multiply by .
Step 4.8.11.1.3
Multiply by .
Step 4.8.11.1.4
Rewrite using the commutative property of multiplication.
Step 4.8.11.1.5
Multiply by by adding the exponents.
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Step 4.8.11.1.5.1
Move .
Step 4.8.11.1.5.2
Use the power rule to combine exponents.
Step 4.8.11.1.5.3
Add and .
Step 4.8.11.1.6
Multiply by .
Step 4.8.11.1.7
Multiply by .
Step 4.8.11.2
Subtract from .
Step 4.8.12
Apply the distributive property.
Step 4.8.13
Simplify.
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Step 4.8.13.1
Multiply by .
Step 4.8.13.2
Multiply by .
Step 4.8.14
Subtract from .
Step 4.8.15
Add and .
Step 4.8.16
Add and .
Step 4.8.17
Rewrite in a factored form.
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Step 4.8.17.1
Factor out of .
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Step 4.8.17.1.1
Factor out of .
Step 4.8.17.1.2
Factor out of .
Step 4.8.17.1.3
Factor out of .
Step 4.8.17.2
Rewrite as .
Step 4.8.17.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.9
Cancel the common factor.
Step 4.10
Rewrite the expression.
Step 4.11
Cancel the common factor.
Step 4.12
Rewrite the expression.