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

Problem

(d(sec(θ))*tan(θ))/d(θ)

Solution

1. Identify the rule needed for the derivative of a product of two functions, which is the product rule: (d(u)*v)/d(θ)=ud(v)/d(θ)+vd(u)/d(θ)

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

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

4. Apply the product rule formula by substituting the functions and their derivatives.

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

5. Simplify the expression by multiplying the terms and factoring out the common term sec(θ)

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

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

Final Answer

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

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