Find the Derivative - d/dx y = natural log of sec(x)+tan(x)

Problem

d()/d(x)*ln(sec(x)+tan(x))

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is ln(u) and the inner function is u=sec(x)+tan(x)

2. Apply the Chain Rule for the natural logarithm, which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x)

3. Differentiate the inner expression sec(x)+tan(x) with respect to x

4. Substitute the derivatives of the trigonometric functions: d(sec(x))/d(x)=sec(x)*tan(x) and d(tan(x))/d(x)=sec2(x)

5. Combine the results into the Chain Rule formula.

d(ln(sec(x)+tan(x)))/d(x)=1/(sec(x)+tan(x))⋅(sec(x)*tan(x)+sec2(x))

6. Factor out the common term sec(x) from the numerator.

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

7. Simplify the expression by canceling the common factor (sec(x)+tan(x)) in the numerator and denominator.

Final Answer

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

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