Calculus Examples

Find the Derivative - d/dx y=xarcsin(x)+ square root of 1-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
Differentiate using the Product Rule which states that is where and .
Step 2.2
The derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Combine and .
Step 2.5
Multiply by .
Step 3
Evaluate .
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Step 3.1
Use to rewrite as .
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
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
Differentiate using the Power Rule which states that is where .
Step 3.7
To write as a fraction with a common denominator, multiply by .
Step 3.8
Combine and .
Step 3.9
Combine the numerators over the common denominator.
Step 3.10
Simplify the numerator.
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Step 3.10.1
Multiply by .
Step 3.10.2
Subtract from .
Step 3.11
Move the negative in front of the fraction.
Step 3.12
Multiply by .
Step 3.13
Subtract from .
Step 3.14
Combine and .
Step 3.15
Combine and .
Step 3.16
Combine and .
Step 3.17
Move to the denominator using the negative exponent rule .
Step 3.18
Factor out of .
Step 3.19
Cancel the common factors.
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Step 3.19.1
Factor out of .
Step 3.19.2
Cancel the common factor.
Step 3.19.3
Rewrite the expression.
Step 3.20
Move the negative in front of the fraction.
Step 4
Simplify.
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Step 4.1
Reorder terms.
Step 4.2
Simplify each term.
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Step 4.2.1
Simplify the denominator.
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Step 4.2.1.1
Rewrite as .
Step 4.2.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.2
Multiply by .
Step 4.2.3
Combine and simplify the denominator.
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Step 4.2.3.1
Multiply by .
Step 4.2.3.2
Raise to the power of .
Step 4.2.3.3
Raise to the power of .
Step 4.2.3.4
Use the power rule to combine exponents.
Step 4.2.3.5
Add and .
Step 4.2.3.6
Rewrite as .
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Step 4.2.3.6.1
Use to rewrite as .
Step 4.2.3.6.2
Apply the power rule and multiply exponents, .
Step 4.2.3.6.3
Combine and .
Step 4.2.3.6.4
Cancel the common factor of .
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Step 4.2.3.6.4.1
Cancel the common factor.
Step 4.2.3.6.4.2
Rewrite the expression.
Step 4.2.3.6.5
Simplify.
Step 4.3
To write as a fraction with a common denominator, multiply by .
Step 4.4
To write as a fraction with a common denominator, multiply by .
Step 4.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.5.1
Multiply by .
Step 4.5.2
Multiply by .
Step 4.5.3
Reorder the factors of .
Step 4.5.4
Reorder the factors of .
Step 4.6
Combine the numerators over the common denominator.
Step 4.7
Simplify the numerator.
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Step 4.7.1
Factor out of .
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Step 4.7.1.1
Factor out of .
Step 4.7.1.2
Factor out of .
Step 4.7.1.3
Factor out of .
Step 4.7.2
Expand using the FOIL Method.
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Step 4.7.2.1
Apply the distributive property.
Step 4.7.2.2
Apply the distributive property.
Step 4.7.2.3
Apply the distributive property.
Step 4.7.3
Simplify and combine like terms.
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Step 4.7.3.1
Simplify each term.
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Step 4.7.3.1.1
Multiply by .
Step 4.7.3.1.2
Multiply by .
Step 4.7.3.1.3
Multiply by .
Step 4.7.3.1.4
Rewrite using the commutative property of multiplication.
Step 4.7.3.1.5
Multiply by by adding the exponents.
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Step 4.7.3.1.5.1
Move .
Step 4.7.3.1.5.2
Multiply by .
Step 4.7.3.2
Add and .
Step 4.7.3.3
Add and .
Step 4.7.4
Apply the distributive property.
Step 4.7.5
Multiply by .
Step 4.7.6
Multiply .
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Step 4.7.6.1
Multiply by .
Step 4.7.6.2
Multiply by .
Step 4.8
To write as a fraction with a common denominator, multiply by .
Step 4.9
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.9.1
Combine and .
Step 4.9.2
Reorder the factors of .
Step 4.10
Combine the numerators over the common denominator.
Step 4.11
Simplify the numerator.
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Step 4.11.1
Use to rewrite as .
Step 4.11.2
Simplify each term.
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Step 4.11.2.1
Expand using the FOIL Method.
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Step 4.11.2.1.1
Apply the distributive property.
Step 4.11.2.1.2
Apply the distributive property.
Step 4.11.2.1.3
Apply the distributive property.
Step 4.11.2.2
Simplify and combine like terms.
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Step 4.11.2.2.1
Simplify each term.
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Step 4.11.2.2.1.1
Multiply by .
Step 4.11.2.2.1.2
Multiply by .
Step 4.11.2.2.1.3
Multiply by .
Step 4.11.2.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.11.2.2.1.5
Multiply by by adding the exponents.
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Step 4.11.2.2.1.5.1
Move .
Step 4.11.2.2.1.5.2
Multiply by .
Step 4.11.2.2.2
Add and .
Step 4.11.2.2.3
Add and .
Step 4.11.2.3
Multiply by by adding the exponents.
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Step 4.11.2.3.1
Use the power rule to combine exponents.
Step 4.11.2.3.2
Combine the numerators over the common denominator.
Step 4.11.2.3.3
Add and .
Step 4.11.2.3.4
Divide by .
Step 4.11.2.4
Simplify .
Step 4.11.3
Combine the opposite terms in .
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Step 4.11.3.1
Subtract from .
Step 4.11.3.2
Add and .
Step 4.11.3.3
Add and .
Step 4.11.4
Multiply by .
Step 4.11.5
Expand using the FOIL Method.
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Step 4.11.5.1
Apply the distributive property.
Step 4.11.5.2
Apply the distributive property.
Step 4.11.5.3
Apply the distributive property.
Step 4.11.6
Simplify and combine like terms.
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Step 4.11.6.1
Simplify each term.
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Step 4.11.6.1.1
Multiply by .
Step 4.11.6.1.2
Multiply by .
Step 4.11.6.1.3
Multiply by .
Step 4.11.6.1.4
Rewrite using the commutative property of multiplication.
Step 4.11.6.1.5
Multiply by by adding the exponents.
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Step 4.11.6.1.5.1
Move .
Step 4.11.6.1.5.2
Multiply by .
Step 4.11.6.2
Add and .
Step 4.11.6.3
Add and .
Step 4.11.7
Multiply by by adding the exponents.
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Step 4.11.7.1
Multiply by .
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Step 4.11.7.1.1
Raise to the power of .
Step 4.11.7.1.2
Use the power rule to combine exponents.
Step 4.11.7.2
Write as a fraction with a common denominator.
Step 4.11.7.3
Combine the numerators over the common denominator.
Step 4.11.7.4
Add and .
Step 4.11.8
Add and .
Step 4.12
Move to the numerator using the negative exponent rule .
Step 4.13
Simplify the numerator.
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Step 4.13.1
Multiply by by adding the exponents.
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Step 4.13.1.1
Move .
Step 4.13.1.2
Use the power rule to combine exponents.
Step 4.13.1.3
Combine the numerators over the common denominator.
Step 4.13.1.4
Add and .
Step 4.13.1.5
Divide by .
Step 4.13.2
Simplify .
Step 4.14
Simplify the numerator.
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Step 4.14.1
Rewrite as .
Step 4.14.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.15
Cancel the common factor of .
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Step 4.15.1
Cancel the common factor.
Step 4.15.2
Rewrite the expression.
Step 4.16
Cancel the common factor of .
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Step 4.16.1
Cancel the common factor.
Step 4.16.2
Divide by .