Mathematics 2011 JAMB Past Questions - Question 37: Find the derivative of sincθ / cosθ

Correct Answer

A

Explanation

Derivative of sin ѳ/cos ѳ = sec² ѳ Sin ѳ /cos ѳ = tan ѳ derivative of tan ѳ = sec² ѳ To find the derivative of sincθ / cosθ, we can use the quotient rule.The quotient rule states that if we have two functions u(θ) and v(θ), the derivative of their quotient is given by:(d/dθ)(u(θ)/v(θ)) = (v(θ)u'(θ) - u(θ)v'(θ)) / (v(θ))^2In this case, u(θ) = sincθ and v(θ) = cosθ.Taking the derivatives of u(θ) and v(θ), we have:u'(θ) = cosθv'(θ) = -sinθNow, we can apply the quotient rule:(d/dθ)((sincθ) / cosθ) = (cosθ * cosθ - sincθ * (-sinθ)) / (cosθ)^2Simplifying:= (cos^2θ + sincθ * sinθ) / cos^2θ= (1 + sincθ * sinθ / cos^2θ)= 1 + tanθ * sincθTherefore, the derivative of sincθ / cosθ is 1 + tanθ * sincθ.