Calculus Examples

Find dy/dx y=(x^2)/( square root of x^2+1)
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
The derivative of with respect to is .
Step 4
Differentiate the right side of the equation.
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Step 4.1
Differentiate using the Quotient Rule which states that is where and .
Step 4.2
Multiply the exponents in .
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Step 4.2.1
Apply the power rule and multiply exponents, .
Step 4.2.2
Cancel the common factor of .
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Step 4.2.2.1
Cancel the common factor.
Step 4.2.2.2
Rewrite the expression.
Step 4.3
Simplify.
Step 4.4
Differentiate using the Power Rule.
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Step 4.4.1
Differentiate using the Power Rule which states that is where .
Step 4.4.2
Move to the left of .
Step 4.5
Differentiate using the chain rule, which states that is where and .
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Step 4.5.1
To apply the Chain Rule, set as .
Step 4.5.2
Differentiate using the Power Rule which states that is where .
Step 4.5.3
Replace all occurrences of with .
Step 4.6
To write as a fraction with a common denominator, multiply by .
Step 4.7
Combine and .
Step 4.8
Combine the numerators over the common denominator.
Step 4.9
Simplify the numerator.
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Step 4.9.1
Multiply by .
Step 4.9.2
Subtract from .
Step 4.10
Combine fractions.
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Step 4.10.1
Move the negative in front of the fraction.
Step 4.10.2
Combine and .
Step 4.10.3
Move to the denominator using the negative exponent rule .
Step 4.10.4
Combine and .
Step 4.11
By the Sum Rule, the derivative of with respect to is .
Step 4.12
Differentiate using the Power Rule which states that is where .
Step 4.13
Since is constant with respect to , the derivative of with respect to is .
Step 4.14
Combine fractions.
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Step 4.14.1
Add and .
Step 4.14.2
Multiply by .
Step 4.14.3
Combine and .
Step 4.14.4
Combine and .
Step 4.15
Raise to the power of .
Step 4.16
Use the power rule to combine exponents.
Step 4.17
Add and .
Step 4.18
Factor out of .
Step 4.19
Cancel the common factors.
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Step 4.19.1
Factor out of .
Step 4.19.2
Cancel the common factor.
Step 4.19.3
Rewrite the expression.
Step 4.20
Move the negative in front of the fraction.
Step 4.21
Combine and using a common denominator.
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Step 4.21.1
Move .
Step 4.21.2
To write as a fraction with a common denominator, multiply by .
Step 4.21.3
Combine the numerators over the common denominator.
Step 4.22
Multiply by by adding the exponents.
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Step 4.22.1
Move .
Step 4.22.2
Use the power rule to combine exponents.
Step 4.22.3
Combine the numerators over the common denominator.
Step 4.22.4
Add and .
Step 4.22.5
Divide by .
Step 4.23
Simplify .
Step 4.24
Rewrite as a product.
Step 4.25
Multiply by .
Step 4.26
Multiply by by adding the exponents.
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Step 4.26.1
Multiply by .
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Step 4.26.1.1
Raise to the power of .
Step 4.26.1.2
Use the power rule to combine exponents.
Step 4.26.2
Write as a fraction with a common denominator.
Step 4.26.3
Combine the numerators over the common denominator.
Step 4.26.4
Add and .
Step 4.27
Simplify.
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Step 4.27.1
Apply the distributive property.
Step 4.27.2
Simplify the numerator.
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Step 4.27.2.1
Simplify each term.
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Step 4.27.2.1.1
Multiply by by adding the exponents.
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Step 4.27.2.1.1.1
Move .
Step 4.27.2.1.1.2
Multiply by .
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Step 4.27.2.1.1.2.1
Raise to the power of .
Step 4.27.2.1.1.2.2
Use the power rule to combine exponents.
Step 4.27.2.1.1.3
Add and .
Step 4.27.2.1.2
Multiply by .
Step 4.27.2.2
Subtract from .
Step 4.27.3
Factor out of .
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Step 4.27.3.1
Factor out of .
Step 4.27.3.2
Factor out of .
Step 4.27.3.3
Factor out of .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .