Find the Derivative - d/dx natural log of sec(7x)+tan(7x)

Problem

d(ln(sec(7*x)+tan(7*x)))/d(x)

Solution

1. Identify the outer function as the natural logarithm and the inner function as u=sec(7*x)+tan(7*x)

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

3. Differentiate the inner expression sec(7*x)+tan(7*x) using the chain rule for trigonometric functions.

4. Calculate the derivative of sec(7*x) which is 7*sec(7*x)*tan(7*x)

5. Calculate the derivative of tan(7*x) which is 7*sec2(7*x)

6. Combine the results into the chain rule formula:

1/(sec(7*x)+tan(7*x))⋅(7*sec(7*x)*tan(7*x)+7*sec2(7*x))

7. Factor out the common term 7*sec(7*x) from the numerator:

(7*sec(7*x)*(tan(7*x)+sec(7*x)))/(sec(7*x)+tan(7*x))

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

Final Answer

d(ln(sec(7*x)+tan(7*x)))/d(x)=7*sec(7*x)

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