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

Problem

d()/d(θ)*sec(θ)*tan(θ)

Solution

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

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

3. Differentiate the individual components: d(sec(θ))/d(θ)=sec(θ)*tan(θ) and d(tan(θ))/d(θ)=sec2(θ)

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

d(y)/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(y)/d(θ)=sec3(θ)+sec(θ)*tan2(θ)

d(y)/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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