Calculus Examples

Find the Derivative - d/dx -cos(x) natural log of sec(x)+tan(x)
Step 1
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Product Rule which states that is where 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
Differentiate using the Sum Rule.
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Step 4.1
Combine and .
Step 4.2
By the Sum Rule, the derivative of with respect to is .
Step 5
The derivative of with respect to is .
Step 6
The derivative of with respect to is .
Step 7
The derivative of with respect to is .
Step 8
Simplify.
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Step 8.1
Apply the distributive property.
Step 8.2
Combine terms.
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Step 8.2.1
Multiply by .
Step 8.2.2
Multiply by .
Step 8.3
Reorder terms.
Step 8.4
Simplify each term.
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Step 8.4.1
Simplify each term.
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Step 8.4.1.1
Rewrite in terms of sines and cosines.
Step 8.4.1.2
Rewrite in terms of sines and cosines.
Step 8.4.1.3
Multiply .
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Step 8.4.1.3.1
Multiply by .
Step 8.4.1.3.2
Raise to the power of .
Step 8.4.1.3.3
Raise to the power of .
Step 8.4.1.3.4
Use the power rule to combine exponents.
Step 8.4.1.3.5
Add and .
Step 8.4.1.4
Rewrite in terms of sines and cosines.
Step 8.4.1.5
Apply the product rule to .
Step 8.4.1.6
One to any power is one.
Step 8.4.2
Apply the distributive property.
Step 8.4.3
Simplify the denominator.
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Step 8.4.3.1
Rewrite in terms of sines and cosines.
Step 8.4.3.2
Rewrite in terms of sines and cosines.
Step 8.4.4
Apply the distributive property.
Step 8.4.5
Cancel the common factor of .
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Step 8.4.5.1
Move the leading negative in into the numerator.
Step 8.4.5.2
Factor out of .
Step 8.4.5.3
Cancel the common factor.
Step 8.4.5.4
Rewrite the expression.
Step 8.4.6
Multiply by .
Step 8.4.7
Cancel the common factor of .
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Step 8.4.7.1
Move the leading negative in into the numerator.
Step 8.4.7.2
Factor out of .
Step 8.4.7.3
Cancel the common factor.
Step 8.4.7.4
Rewrite the expression.
Step 8.4.8
Multiply by .
Step 8.4.9
Simplify each term.
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Step 8.4.9.1
Move the negative in front of the fraction.
Step 8.4.9.2
Move the negative in front of the fraction.
Step 8.4.10
Simplify each term.
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Step 8.4.10.1
Rewrite in terms of sines and cosines.
Step 8.4.10.2
Rewrite in terms of sines and cosines.
Step 8.5
To write as a fraction with a common denominator, multiply by .
Step 8.6
Combine and .
Step 8.7
Combine the numerators over the common denominator.
Step 8.8
Simplify the numerator.
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Step 8.8.1
Factor out of .
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Step 8.8.1.1
Factor out of .
Step 8.8.1.2
Factor out of .
Step 8.8.1.3
Factor out of .
Step 8.8.2
Apply the distributive property.
Step 8.8.3
Cancel the common factor of .
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Step 8.8.3.1
Factor out of .
Step 8.8.3.2
Cancel the common factor.
Step 8.8.3.3
Rewrite the expression.
Step 8.8.4
Cancel the common factor of .
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Step 8.8.4.1
Factor out of .
Step 8.8.4.2
Cancel the common factor.
Step 8.8.4.3
Rewrite the expression.
Step 8.9
Combine the numerators over the common denominator.
Step 8.10
Simplify the numerator.
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Step 8.10.1
Apply the distributive property.
Step 8.10.2
Simplify.
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Step 8.10.2.1
Move to the left of .
Step 8.10.2.2
Multiply .
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Step 8.10.2.2.1
Raise to the power of .
Step 8.10.2.2.2
Raise to the power of .
Step 8.10.2.2.3
Use the power rule to combine exponents.
Step 8.10.2.2.4
Add and .
Step 8.10.3
Rewrite as .
Step 8.10.4
Rewrite in a factored form.
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Step 8.10.4.1
Reorder terms.
Step 8.10.4.2
Factor out the greatest common factor from each group.
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Step 8.10.4.2.1
Group the first two terms and the last two terms.
Step 8.10.4.2.2
Factor out the greatest common factor (GCF) from each group.
Step 8.10.4.3
Factor the polynomial by factoring out the greatest common factor, .
Step 8.11
Simplify each term.
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Step 8.11.1
Convert from to .
Step 8.11.2
Convert from to .
Step 8.12
Simplify each term.
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Step 8.12.1
Convert from to .
Step 8.12.2
Convert from to .