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