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Calculus Examples
ln(sec(x))ln(sec(x))
Step 1
Step 1.1
To apply the Chain Rule, set uu as sec(x)sec(x).
ddu[ln(u)]ddx[sec(x)]ddu[ln(u)]ddx[sec(x)]
Step 1.2
The derivative of ln(u)ln(u) with respect to uu is 1u1u.
1uddx[sec(x)]1uddx[sec(x)]
Step 1.3
Replace all occurrences of uu with sec(x)sec(x).
1sec(x)ddx[sec(x)]1sec(x)ddx[sec(x)]
1sec(x)ddx[sec(x)]1sec(x)ddx[sec(x)]
Step 2
Rewrite sec(x)sec(x) in terms of sines and cosines.
11cos(x)ddx[sec(x)]11cos(x)ddx[sec(x)]
Step 3
Multiply by the reciprocal of the fraction to divide by 1cos(x)1cos(x).
1cos(x)ddx[sec(x)]1cos(x)ddx[sec(x)]
Step 4
Multiply cos(x)cos(x) by 11.
cos(x)ddx[sec(x)]cos(x)ddx[sec(x)]
Step 5
The derivative of sec(x)sec(x) with respect to xx is sec(x)tan(x)sec(x)tan(x).
cos(x)(sec(x)tan(x))cos(x)(sec(x)tan(x))
Step 6
Step 6.1
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 6.1.1
Reorder cos(x)cos(x) and sec(x)sec(x).
sec(x)cos(x)tan(x)sec(x)cos(x)tan(x)
Step 6.1.2
Rewrite cos(x)sec(x)cos(x)sec(x) in terms of sines and cosines.
1cos(x)cos(x)tan(x)1cos(x)cos(x)tan(x)
Step 6.1.3
Cancel the common factors.
1tan(x)1tan(x)
1tan(x)1tan(x)
Step 6.2
Multiply tan(x)tan(x) by 11.
tan(x)tan(x)
Step 6.3
Rewrite tan(x)tan(x) in terms of sines and cosines.
sin(x)cos(x)sin(x)cos(x)
Step 6.4
Convert from sin(x)cos(x)sin(x)cos(x) to tan(x)tan(x).
tan(x)tan(x)
tan(x)tan(x)