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

Problem

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

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=sec(x) and v(x)=tan(x)

2. Apply the product rule, which states that (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the individual trigonometric functions: d(sec(x))/d(x)=sec(x)*tan(x) and d(tan(x))/d(x)=sec2(x)

4. Substitute these derivatives back into the product rule formula.

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

5. Simplify the expression by multiplying the terms.

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

6. Factor out the common term sec(x) if desired.

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

Final Answer

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

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