Calculus Examples

Find dy/dx y=1/12*((sec(3x)^2)(tan(3x)^2-1))
Step 1
Rewrite the right side with rational exponents.
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Step 1.1
Multiply by .
Step 1.2
Remove parentheses.
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
Combine and .
Step 4.2
Differentiate using the Product Rule which states that is where and .
Step 4.3
By the Sum Rule, the derivative of with respect to is .
Step 4.4
Differentiate using the chain rule, which states that is where and .
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Step 4.4.1
To apply the Chain Rule, set as .
Step 4.4.2
Differentiate using the Power Rule which states that is where .
Step 4.4.3
Replace all occurrences of with .
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
The derivative of with respect to is .
Step 4.5.3
Replace all occurrences of with .
Step 4.6
Differentiate.
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Step 4.6.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.6.2
Multiply by .
Step 4.6.3
Differentiate using the Power Rule which states that is where .
Step 4.6.4
Multiply by .
Step 4.6.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.6.6
Combine fractions.
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Step 4.6.6.1
Add and .
Step 4.6.6.2
Combine and .
Step 4.6.6.3
Combine and .
Step 4.6.6.4
Combine and .
Step 4.7
Multiply by by adding the exponents.
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Step 4.7.1
Move .
Step 4.7.2
Use the power rule to combine exponents.
Step 4.7.3
Add and .
Step 4.8
Differentiate using the Constant Multiple Rule.
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Step 4.8.1
Move to the left of .
Step 4.8.2
Cancel the common factor of and .
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Step 4.8.2.1
Factor out of .
Step 4.8.2.2
Cancel the common factors.
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Step 4.8.2.2.1
Factor out of .
Step 4.8.2.2.2
Cancel the common factor.
Step 4.8.2.2.3
Rewrite the expression.
Step 4.8.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.9
Differentiate using the chain rule, which states that is where and .
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Step 4.9.1
To apply the Chain Rule, set as .
Step 4.9.2
Differentiate using the Power Rule which states that is where .
Step 4.9.3
Replace all occurrences of with .
Step 4.10
Simplify terms.
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Step 4.10.1
Combine and .
Step 4.10.2
Combine and .
Step 4.10.3
Move to the left of .
Step 4.10.4
Cancel the common factor of and .
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Step 4.10.4.1
Factor out of .
Step 4.10.4.2
Cancel the common factors.
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Step 4.10.4.2.1
Factor out of .
Step 4.10.4.2.2
Cancel the common factor.
Step 4.10.4.2.3
Rewrite the expression.
Step 4.11
Differentiate using the chain rule, which states that is where and .
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Step 4.11.1
To apply the Chain Rule, set as .
Step 4.11.2
The derivative of with respect to is .
Step 4.11.3
Replace all occurrences of with .
Step 4.12
Combine and .
Step 4.13
Raise to the power of .
Step 4.14
Raise to the power of .
Step 4.15
Use the power rule to combine exponents.
Step 4.16
Combine fractions.
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Step 4.16.1
Add and .
Step 4.16.2
Combine and .
Step 4.17
Since is constant with respect to , the derivative of with respect to is .
Step 4.18
Simplify terms.
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Step 4.18.1
Combine and .
Step 4.18.2
Cancel the common factors.
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Step 4.18.2.1
Factor out of .
Step 4.18.2.2
Cancel the common factor.
Step 4.18.2.3
Rewrite the expression.
Step 4.19
Differentiate using the Power Rule which states that is where .
Step 4.20
Multiply by .
Step 4.21
To write as a fraction with a common denominator, multiply by .
Step 4.22
Combine and .
Step 4.23
Combine the numerators over the common denominator.
Step 4.24
Combine and .
Step 4.25
Cancel the common factor of .
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Step 4.25.1
Cancel the common factor.
Step 4.25.2
Divide by .
Step 4.26
Simplify.
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Step 4.26.1
Apply the distributive property.
Step 4.26.2
Apply the distributive property.
Step 4.26.3
Simplify the numerator.
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Step 4.26.3.1
Factor out of .
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Step 4.26.3.1.1
Factor out of .
Step 4.26.3.1.2
Factor out of .
Step 4.26.3.1.3
Factor out of .
Step 4.26.3.1.4
Factor out of .
Step 4.26.3.1.5
Factor out of .
Step 4.26.3.2
Move .
Step 4.26.3.3
Apply pythagorean identity.
Step 4.26.3.4
Multiply .
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Step 4.26.3.4.1
Raise to the power of .
Step 4.26.3.4.2
Raise to the power of .
Step 4.26.3.4.3
Use the power rule to combine exponents.
Step 4.26.3.4.4
Add and .
Step 4.26.3.5
Add and .
Step 4.26.3.6
Rewrite using the commutative property of multiplication.
Step 4.26.3.7
Multiply by by adding the exponents.
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Step 4.26.3.7.1
Move .
Step 4.26.3.7.2
Multiply by .
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Step 4.26.3.7.2.1
Raise to the power of .
Step 4.26.3.7.2.2
Use the power rule to combine exponents.
Step 4.26.3.7.3
Add and .
Step 4.26.4
Cancel the common factor of .
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Step 4.26.4.1
Cancel the common factor.
Step 4.26.4.2
Divide by .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .