Find the Derivative - d/dx natural log of sec(x)

Problem

d(ln(sec(x)))/d(x)

Solution

1. Identify the outer and inner functions for the chain rule, where the outer function is ln(u) and the inner function is u=sec(x)

2. Apply the chain rule by differentiating the natural log with respect to its argument and then multiplying by the derivative of the inner function.

d(ln(sec(x)))/d(x)=1/sec(x)⋅d(sec(x))/d(x)

3. Differentiate the inner function sec(x) which results in sec(x)*tan(x)

d(ln(sec(x)))/d(x)=1/sec(x)⋅sec(x)*tan(x)

4. Simplify the expression by canceling the sec(x) terms in the numerator and denominator.

d(ln(sec(x)))/d(x)=tan(x)

Final Answer

d(ln(sec(x)))/d(x)=tan(x)

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