Calculus Examples

Find the Derivative - d/dx y=(7-sec(x))/(tan(x))
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
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
Add and .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 3
The derivative of with respect to is .
Step 4
Raise to the power of .
Step 5
Raise to the power of .
Step 6
Use the power rule to combine exponents.
Step 7
Add and .
Step 8
The derivative of with respect to is .
Step 9
Simplify.
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Step 9.1
Apply the distributive property.
Step 9.2
Apply the distributive property.
Step 9.3
Simplify the numerator.
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Step 9.3.1
Simplify each term.
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Step 9.3.1.1
Multiply by .
Step 9.3.1.2
Multiply by by adding the exponents.
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Step 9.3.1.2.1
Move .
Step 9.3.1.2.2
Multiply by .
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Step 9.3.1.2.2.1
Raise to the power of .
Step 9.3.1.2.2.2
Use the power rule to combine exponents.
Step 9.3.1.2.3
Add and .
Step 9.3.1.3
Multiply .
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Step 9.3.1.3.1
Multiply by .
Step 9.3.1.3.2
Multiply by .
Step 9.3.2
Factor out of .
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Step 9.3.2.1
Factor out of .
Step 9.3.2.2
Factor out of .
Step 9.3.2.3
Factor out of .
Step 9.3.2.4
Factor out of .
Step 9.3.2.5
Factor out of .
Step 9.3.3
Move .
Step 9.3.4
Reorder and .
Step 9.3.5
Apply pythagorean identity.
Step 9.3.6
Apply the distributive property.
Step 9.3.7
Multiply by .
Step 9.3.8
Rewrite using the commutative property of multiplication.
Step 9.3.9
Multiply .
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Step 9.3.9.1
Raise to the power of .
Step 9.3.9.2
Raise to the power of .
Step 9.3.9.3
Use the power rule to combine exponents.
Step 9.3.9.4
Add and .
Step 9.4
Factor out of .
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Step 9.4.1
Multiply by .
Step 9.4.2
Factor out of .
Step 9.4.3
Factor out of .
Step 9.5
Factor out of .
Step 9.6
Separate fractions.
Step 9.7
Rewrite in terms of sines and cosines.
Step 9.8
Rewrite in terms of sines and cosines.
Step 9.9
Multiply by the reciprocal of the fraction to divide by .
Step 9.10
Cancel the common factor of .
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Step 9.10.1
Cancel the common factor.
Step 9.10.2
Rewrite the expression.
Step 9.11
Convert from to .
Step 9.12
Combine and .
Step 9.13
Separate fractions.
Step 9.14
Rewrite in terms of sines and cosines.
Step 9.15
Rewrite in terms of sines and cosines.
Step 9.16
Multiply by the reciprocal of the fraction to divide by .
Step 9.17
Simplify.
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Step 9.17.1
Convert from to .
Step 9.17.2
Convert from to .
Step 9.18
Divide by .
Step 9.19
Apply the distributive property.
Step 9.20
Multiply by .