Calculus Examples

Find the Derivative - d/d@VAR f(x)=(sec(x))/(2-cos(x))
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
The derivative of with respect to is .
Step 3
Differentiate.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Add and .
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5
Multiply.
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Step 3.5.1
Multiply by .
Step 3.5.2
Multiply by .
Step 4
The derivative of with respect to is .
Step 5
Simplify.
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Step 5.1
Apply the distributive property.
Step 5.2
Apply the distributive property.
Step 5.3
Simplify the numerator.
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Step 5.3.1
Simplify each term.
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Step 5.3.1.1
Rewrite in terms of sines and cosines.
Step 5.3.1.2
Combine and .
Step 5.3.1.3
Rewrite in terms of sines and cosines.
Step 5.3.1.4
Multiply .
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Step 5.3.1.4.1
Multiply by .
Step 5.3.1.4.2
Raise to the power of .
Step 5.3.1.4.3
Raise to the power of .
Step 5.3.1.4.4
Use the power rule to combine exponents.
Step 5.3.1.4.5
Add and .
Step 5.3.1.5
Rewrite in terms of sines and cosines, then cancel the common factors.
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Step 5.3.1.5.1
Add parentheses.
Step 5.3.1.5.2
Reorder and .
Step 5.3.1.5.3
Rewrite in terms of sines and cosines.
Step 5.3.1.5.4
Cancel the common factors.
Step 5.3.1.6
Multiply by .
Step 5.3.1.7
Rewrite in terms of sines and cosines.
Step 5.3.1.8
Rewrite as .
Step 5.3.1.9
Rewrite using the commutative property of multiplication.
Step 5.3.1.10
Rewrite in terms of sines and cosines.
Step 5.3.1.11
Combine and .
Step 5.3.2
Combine the numerators over the common denominator.
Step 5.3.3
Subtract from .
Step 5.3.4
Move the negative in front of the fraction.
Step 5.3.5
Simplify each term.
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Step 5.3.5.1
Factor out of .
Step 5.3.5.2
Separate fractions.
Step 5.3.5.3
Convert from to .
Step 5.3.5.4
Separate fractions.
Step 5.3.5.5
Convert from to .
Step 5.3.5.6
Divide by .
Step 5.3.5.7
Separate fractions.
Step 5.3.5.8
Convert from to .
Step 5.3.5.9
Divide by .
Step 5.3.5.10
Multiply by .
Step 5.4
Simplify the numerator.
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Step 5.4.1
Factor out of .
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Step 5.4.1.1
Factor out of .
Step 5.4.1.2
Factor out of .
Step 5.4.1.3
Factor out of .
Step 5.4.2
Rewrite in terms of sines and cosines.
Step 5.4.3
Rewrite in terms of sines and cosines.
Step 5.4.4
Combine and .
Step 5.5
Factor out of .
Step 5.6
Multiply by .
Step 5.7
Separate fractions.
Step 5.8
Convert from to .
Step 5.9
Convert from to .
Step 5.10
Combine and .
Step 5.11
Reorder factors in .