Calculus Examples

Find the Derivative - d/dx y=(x^3-2)/(sec(4x^2))
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Add and .
Step 3
Differentiate using the chain rule, which states that is where and .
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Step 3.1
To apply the Chain Rule, set as .
Step 3.2
The derivative of with respect to is .
Step 3.3
Replace all occurrences of with .
Step 4
Factor out of .
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Step 4.1
Factor out of .
Step 4.2
Factor out of .
Step 5
Cancel the common factors.
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Step 5.1
Factor out of .
Step 5.2
Cancel the common factor.
Step 5.3
Rewrite the expression.
Step 6
Since is constant with respect to , the derivative of with respect to is .
Step 7
Multiply by .
Step 8
Differentiate using the Power Rule which states that is where .
Step 9
Multiply by .
Step 10
Simplify.
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Step 10.1
Apply the distributive property.
Step 10.2
Apply the distributive property.
Step 10.3
Simplify the numerator.
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Step 10.3.1
Simplify each term.
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Step 10.3.1.1
Multiply by by adding the exponents.
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Step 10.3.1.1.1
Move .
Step 10.3.1.1.2
Multiply by .
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Step 10.3.1.1.2.1
Raise to the power of .
Step 10.3.1.1.2.2
Use the power rule to combine exponents.
Step 10.3.1.1.3
Add and .
Step 10.3.1.2
Multiply by .
Step 10.3.2
Reorder factors in .