Calculus Examples

Find the Derivative Using Quotient Rule - d/dx integral of (sec(x)+csc(x)+cot(x))/(tan(x)^2+sin(x)^2+cos(x)^2) with respect to x
sec(x)+csc(x)+cot(x)tan2(x)+sin2(x)+cos2(x)dxsec(x)+csc(x)+cot(x)tan2(x)+sin2(x)+cos2(x)dx
Step 1
This derivative could not be completed using the quotient rule. Mathway will use another method.
Step 2
sec(x)+csc(x)+cot(x)tan2(x)+sin2(x)+cos2(x)dx is an antiderivative of sec(x)+csc(x)+cot(x)tan2(x)+sin2(x)+cos2(x), so by definition ddx[sec(x)+csc(x)+cot(x)tan2(x)+sin2(x)+cos2(x)dx] is sec(x)+csc(x)+cot(x)tan2(x)+sin2(x)+cos2(x).
sec(x)+csc(x)+cot(x)tan2(x)+sin2(x)+cos2(x)
Step 3
Simplify.
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Step 3.1
Apply pythagorean identity.
sec(x)+csc(x)+cot(x)tan2(x)+1
Step 3.2
Apply pythagorean identity.
sec(x)+csc(x)+cot(x)sec2(x)
Step 3.3
Simplify the numerator.
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Step 3.3.1
Rewrite sec(x) in terms of sines and cosines.
1cos(x)+csc(x)+cot(x)sec2(x)
Step 3.3.2
Rewrite csc(x) in terms of sines and cosines.
1cos(x)+1sin(x)+cot(x)sec2(x)
Step 3.3.3
Rewrite cot(x) in terms of sines and cosines.
1cos(x)+1sin(x)+cos(x)sin(x)sec2(x)
1cos(x)+1sin(x)+cos(x)sin(x)sec2(x)
Step 3.4
Simplify the denominator.
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Step 3.4.1
Rewrite sec(x) in terms of sines and cosines.
1cos(x)+1sin(x)+cos(x)sin(x)(1cos(x))2
Step 3.4.2
Apply the product rule to 1cos(x).
1cos(x)+1sin(x)+cos(x)sin(x)12cos2(x)
Step 3.4.3
One to any power is one.
1cos(x)+1sin(x)+cos(x)sin(x)1cos2(x)
1cos(x)+1sin(x)+cos(x)sin(x)1cos2(x)
Step 3.5
Multiply the numerator by the reciprocal of the denominator.
(1cos(x)+1sin(x)+cos(x)sin(x))cos2(x)
Step 3.6
Apply the distributive property.
1cos(x)cos2(x)+1sin(x)cos2(x)+cos(x)sin(x)cos2(x)
Step 3.7
Simplify.
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Step 3.7.1
Cancel the common factor of cos(x).
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Step 3.7.1.1
Factor cos(x) out of cos2(x).
1cos(x)(cos(x)cos(x))+1sin(x)cos2(x)+cos(x)sin(x)cos2(x)
Step 3.7.1.2
Cancel the common factor.
1cos(x)(cos(x)cos(x))+1sin(x)cos2(x)+cos(x)sin(x)cos2(x)
Step 3.7.1.3
Rewrite the expression.
cos(x)+1sin(x)cos2(x)+cos(x)sin(x)cos2(x)
cos(x)+1sin(x)cos2(x)+cos(x)sin(x)cos2(x)
Step 3.7.2
Combine 1sin(x) and cos2(x).
cos(x)+cos2(x)sin(x)+cos(x)sin(x)cos2(x)
Step 3.7.3
Multiply cos(x)sin(x)cos2(x).
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Step 3.7.3.1
Combine cos(x)sin(x) and cos2(x).
cos(x)+cos2(x)sin(x)+cos(x)cos2(x)sin(x)
Step 3.7.3.2
Multiply cos(x) by cos2(x) by adding the exponents.
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Step 3.7.3.2.1
Multiply cos(x) by cos2(x).
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Step 3.7.3.2.1.1
Raise cos(x) to the power of 1.
cos(x)+cos2(x)sin(x)+cos1(x)cos2(x)sin(x)
Step 3.7.3.2.1.2
Use the power rule aman=am+n to combine exponents.
cos(x)+cos2(x)sin(x)+cos(x)1+2sin(x)
cos(x)+cos2(x)sin(x)+cos(x)1+2sin(x)
Step 3.7.3.2.2
Add 1 and 2.
cos(x)+cos2(x)sin(x)+cos3(x)sin(x)
cos(x)+cos2(x)sin(x)+cos3(x)sin(x)
cos(x)+cos2(x)sin(x)+cos3(x)sin(x)
cos(x)+cos2(x)sin(x)+cos3(x)sin(x)
Step 3.8
Simplify each term.
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Step 3.8.1
Factor cos(x) out of cos2(x).
cos(x)+cos(x)cos(x)sin(x)+cos3(x)sin(x)
Step 3.8.2
Separate fractions.
cos(x)+cos(x)1cos(x)sin(x)+cos3(x)sin(x)
Step 3.8.3
Convert from cos(x)sin(x) to cot(x).
cos(x)+cos(x)1cot(x)+cos3(x)sin(x)
Step 3.8.4
Divide cos(x) by 1.
cos(x)+cos(x)cot(x)+cos3(x)sin(x)
Step 3.8.5
Factor cos(x) out of cos3(x).
cos(x)+cos(x)cot(x)+cos(x)cos2(x)sin(x)
Step 3.8.6
Separate fractions.
cos(x)+cos(x)cot(x)+cos2(x)1cos(x)sin(x)
Step 3.8.7
Convert from cos(x)sin(x) to cot(x).
cos(x)+cos(x)cot(x)+cos2(x)1cot(x)
Step 3.8.8
Divide cos2(x) by 1.
cos(x)+cos(x)cot(x)+cos2(x)cot(x)
cos(x)+cos(x)cot(x)+cos2(x)cot(x)
cos(x)+cos(x)cot(x)+cos2(x)cot(x)
 [x2  12  π  xdx ]