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

Problem

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

Solution

1. Identify the product rule for differentiation, which states that for two functions u and v the derivative is ud(v)/d(x)+vd(u)/d(x)

2. Assign the functions u=tan(x) and v=sec(x)

3. Differentiate each component: d(tan(x))/d(x)=sec2(x) and d(sec(x))/d(x)=sec(x)*tan(x)

4. Apply the product rule formula:

(d(tan(x))*sec(x))/d(x)=tan(x)*(sec(x)*tan(x))+sec(x)*(sec2(x))

5. Simplify the expression by multiplying the terms:

(d(tan(x))*sec(x))/d(x)=sec(x)*tan2(x)+sec3(x)

6. Factor out the common term sec(x)

(d(tan(x))*sec(x))/d(x)=sec(x)*(tan2(x)+sec2(x))

Final Answer

(d(tan(x))*sec(x))/d(x)=sec(x)*(tan2(x)+sec2(x))

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