Calculus Examples

Find the Derivative - d/dx y = natural log of (x^4+5x^2)^(3/2)
Step 1
Differentiate using the chain rule, which states that is where and .
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Step 1.1
To apply the Chain Rule, set as .
Step 1.2
The derivative of with respect to is .
Step 1.3
Replace all occurrences of with .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Replace all occurrences of with .
Step 3
To write as a fraction with a common denominator, multiply by .
Step 4
Combine and .
Step 5
Combine the numerators over the common denominator.
Step 6
Simplify the numerator.
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Step 6.1
Multiply by .
Step 6.2
Subtract from .
Step 7
Combine and .
Step 8
Multiply by .
Step 9
Move to the denominator using the negative exponent rule .
Step 10
Simplify the denominator.
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Step 10.1
Multiply by by adding the exponents.
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Step 10.1.1
Move .
Step 10.1.2
Use the power rule to combine exponents.
Step 10.1.3
Combine the numerators over the common denominator.
Step 10.1.4
Add and .
Step 10.1.5
Divide by .
Step 10.2
Simplify .
Step 11
By the Sum Rule, the derivative of with respect to is .
Step 12
Differentiate using the Power Rule which states that is where .
Step 13
Since is constant with respect to , the derivative of with respect to is .
Step 14
Differentiate using the Power Rule which states that is where .
Step 15
Multiply by .
Step 16
Simplify.
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Step 16.1
Apply the distributive property.
Step 16.2
Multiply by .
Step 16.3
Reorder the factors of .
Step 16.4
Factor out of .
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Step 16.4.1
Factor out of .
Step 16.4.2
Factor out of .
Step 16.4.3
Factor out of .
Step 16.5
Multiply by .
Step 16.6
Simplify the numerator.
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Step 16.6.1
Factor out of .
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Step 16.6.1.1
Factor out of .
Step 16.6.1.2
Factor out of .
Step 16.6.1.3
Factor out of .
Step 16.6.2
Multiply by .
Step 16.7
Cancel the common factor of and .
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Step 16.7.1
Factor out of .
Step 16.7.2
Cancel the common factors.
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Step 16.7.2.1
Factor out of .
Step 16.7.2.2
Cancel the common factor.
Step 16.7.2.3
Rewrite the expression.
Step 16.8
Cancel the common factor of and .
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Step 16.8.1
Factor out of .
Step 16.8.2
Cancel the common factors.
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Step 16.8.2.1
Factor out of .
Step 16.8.2.2
Cancel the common factor.
Step 16.8.2.3
Rewrite the expression.