Find the Derivative - d/dx sec(tan(x))

Problem

d(sec(tan(x)))/d(x)

Solution

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

2. Apply the Chain Rule which states that (d(ƒ)*(g(x)))/d(x)=ƒ^′*(g(x))⋅g(x)^′

3. Differentiate the outer function sec(u) with respect to u The derivative of sec(u) is sec(u)*tan(u)

4. Substitute u=tan(x) into the derivative of the outer function.

sec(tan(x))*tan(tan(x))

5. Differentiate the inner function tan(x) with respect to x The derivative of tan(x) is sec^2(x)

6. Multiply the results together to find the final derivative.

sec(tan(x))*tan(tan(x))⋅sec^2(x)

Final Answer

d(sec(tan(x)))/d(x)=sec^2(x)*sec(tan(x))*tan(tan(x))

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