Calculus Examples

Find the Derivative - d/dx f(x)=(550x^2)/( square root of x^2+12)
Step 1
Differentiate using the Constant Multiple Rule.
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Step 1.1
Use to rewrite as .
Step 1.2
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Multiply the exponents in .
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Step 3.1
Apply the power rule and multiply exponents, .
Step 3.2
Cancel the common factor of .
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Step 3.2.1
Cancel the common factor.
Step 3.2.2
Rewrite the expression.
Step 4
Simplify.
Step 5
Differentiate using the Power Rule.
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Step 5.1
Differentiate using the Power Rule which states that is where .
Step 5.2
Move to the left of .
Step 6
Differentiate using the chain rule, which states that is where and .
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Step 6.1
To apply the Chain Rule, set as .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Replace all occurrences of with .
Step 7
To write as a fraction with a common denominator, multiply by .
Step 8
Combine and .
Step 9
Combine the numerators over the common denominator.
Step 10
Simplify the numerator.
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Step 10.1
Multiply by .
Step 10.2
Subtract from .
Step 11
Combine fractions.
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Step 11.1
Move the negative in front of the fraction.
Step 11.2
Combine and .
Step 11.3
Move to the denominator using the negative exponent rule .
Step 11.4
Combine and .
Step 12
By the Sum Rule, the derivative of with respect to is .
Step 13
Differentiate using the Power Rule which states that is where .
Step 14
Since is constant with respect to , the derivative of with respect to is .
Step 15
Combine fractions.
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Step 15.1
Add and .
Step 15.2
Multiply by .
Step 15.3
Combine and .
Step 15.4
Combine and .
Step 16
Raise to the power of .
Step 17
Use the power rule to combine exponents.
Step 18
Add and .
Step 19
Factor out of .
Step 20
Cancel the common factors.
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Step 20.1
Factor out of .
Step 20.2
Cancel the common factor.
Step 20.3
Rewrite the expression.
Step 21
Move the negative in front of the fraction.
Step 22
Combine and using a common denominator.
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Step 22.1
Move .
Step 22.2
To write as a fraction with a common denominator, multiply by .
Step 22.3
Combine the numerators over the common denominator.
Step 23
Multiply by by adding the exponents.
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Step 23.1
Move .
Step 23.2
Use the power rule to combine exponents.
Step 23.3
Combine the numerators over the common denominator.
Step 23.4
Add and .
Step 23.5
Divide by .
Step 24
Simplify .
Step 25
Rewrite as a product.
Step 26
Multiply by .
Step 27
Multiply by by adding the exponents.
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Step 27.1
Multiply by .
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Step 27.1.1
Raise to the power of .
Step 27.1.2
Use the power rule to combine exponents.
Step 27.2
Write as a fraction with a common denominator.
Step 27.3
Combine the numerators over the common denominator.
Step 27.4
Add and .
Step 28
Combine and .
Step 29
Simplify.
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Step 29.1
Apply the distributive property.
Step 29.2
Apply the distributive property.
Step 29.3
Simplify the numerator.
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Step 29.3.1
Simplify each term.
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Step 29.3.1.1
Multiply by by adding the exponents.
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Step 29.3.1.1.1
Move .
Step 29.3.1.1.2
Multiply by .
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Step 29.3.1.1.2.1
Raise to the power of .
Step 29.3.1.1.2.2
Use the power rule to combine exponents.
Step 29.3.1.1.3
Add and .
Step 29.3.1.2
Multiply by .
Step 29.3.1.3
Multiply by .
Step 29.3.1.4
Multiply by .
Step 29.3.1.5
Multiply by .
Step 29.3.2
Subtract from .
Step 29.4
Factor out of .
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Step 29.4.1
Factor out of .
Step 29.4.2
Factor out of .
Step 29.4.3
Factor out of .