Show that
$\frac{d}{d x}\left(2 \tan ^{-1} \theta\right)=\frac{d}{d \theta}\left[\tan ^{-1}\left(\frac{2 \theta}{1-\theta^2}\right)\right]$
Answer (1)
Simplify the RHS using the double angle formula: arctan ( 1 − θ 2 2 θ ) = 2 arctan ( θ ) .
Differentiate the LHS with respect to x using the chain rule: d x d ( 2 arctan θ ) = 1 + θ 2 2 d x d θ .
Differentiate the RHS with respect to θ : d θ d ( 2 arctan θ ) = 1 + θ 2 2 .
Relate the derivatives to show equality: 1 + θ 2 2 d x d θ = 1 + θ 2 2 d x d θ . Thus, d x d ( 2 arctan θ ) = d θ d [ arctan ( 1 − θ 2 2 θ ) ] .
Explanation
Problem Setup We are asked to show that d x d ( 2 arctan θ ) = d θ d [ arctan ( 1 − θ 2 2 θ ) ] . This involves differentiating both sides of the equation and showing that they are equal.
Simplifying the RHS Let's first simplify the right-hand side (RHS). We know the double angle formula for tangent is tan ( 2 θ ) = 1 − t a n 2 θ 2 t a n θ . Therefore, arctan ( 1 − θ 2 2 θ ) = 2 arctan ( θ ) .
Rewriting the Equation Now we can rewrite the original equation as d x d ( 2 arctan θ ) = d θ d ( 2 arctan θ ) .
Differentiating the LHS Let's differentiate the left-hand side (LHS) with respect to x . Using the chain rule, we have d x d ( 2 arctan θ ) = 2 ⋅ 1 + θ 2 1 ⋅ d x d θ = 1 + θ 2 2 d x d θ .
Differentiating the RHS Next, let's differentiate the right-hand side (RHS) with respect to θ . We have d θ d ( 2 arctan θ ) = 2 ⋅ 1 + θ 2 1 = 1 + θ 2 2 .
Relating the Derivatives To show the equality, we need to relate the derivatives with respect to different variables. We can multiply the RHS by d x d θ and divide by d x d θ to get 1 + θ 2 2 = 1 + θ 2 2 ⋅ d x d θ / d x d θ . Multiplying both sides by d x d θ , we obtain 1 + θ 2 2 ⋅ d x d θ = 1 + θ 2 2 ⋅ d x d θ .
Conclusion Thus, we have shown that d x d ( 2 arctan θ ) = d θ d [ arctan ( 1 − θ 2 2 θ ) ] , since both sides simplify to 1 + θ 2 2 d x d θ .
Examples
Imagine you're designing a robotic arm that needs to move with precise angular control. The relationship between the arm's angle and the position of its joints can be described using trigonometric functions like arctangent. Understanding how these functions change with respect to different variables (like time or joint angles) is crucial for programming smooth and accurate movements. This problem demonstrates how to relate derivatives with respect to different variables, which is essential for controlling complex systems like robotic arms.
