Prove the following: (i) (cot^-1)^' y=-(1)/(1+y^2) for all y ∈ℝ, (ii) (csc^-1)^' y=-(1)/(|y| √(y

step 1 So in the given question we will start from the left hand side. So Cossack square y minus plus C

Prove the following: (i) (cot^-1)^' y=-(1)/(1+y^2) for all y...

Prove the following: (i) $\left(\cot ^{-1}\right)^{\prime} y=-\frac{1}{1+y^{2}}$ for all $y \in \mathbb{R}$, (ii) $\left(\mathrm{csc}^{-1}\right)^{\prime} y=-\frac{1}{|y| \sqrt{y^{2}-1}}$ for all $y \in \mathbb{R}$ with $|y|>1$, (iii) $\left(\mathrm{sec}^{-1}\right)^{\prime} y=\frac{1}{|y| \sqrt{y^{2}-1}}$ for all $y \in \mathbb{R}$ with $|y|>1$.

Prove the following:
(i) $\left(\cot ^{-1}\right)^{\prime} y=-\frac{1}{1+y^{2}}$ for all $y \in \mathbb{R}$,
(ii) $\left(\mathrm{csc}^{-1}\right)^{\prime} y=-\frac{1}{|y| \sqrt{y^{2}-1}}$ for all $y \in \mathbb{R}$ with $|y|>1$,
(iii) $\left(\mathrm{sec}^{-1}\right)^{\prime} y=\frac{1}{|y| \sqrt{y^{2}-1}}$ for all $y \in \mathbb{R}$ with $|y|>1$.

Key Concepts

Implicit Differentiation

This concept involves differentiating both sides of an equation that defines a function implicitly rather than explicitly. When dealing with inverse functions, such as the inverse trigonometric functions, implicit differentiation allows the derivative of the inverse to be determined by relating it back to the original trigonometric function and using the derivative of that function.

Chain Rule

The chain rule is a fundamental technique used to differentiate composite functions. In the context of finding the derivative of an inverse function, the chain rule helps to account for the effect of the inner function within the composite by taking the derivative of the outer function and then multiplying it by the derivative of the inner function.

Differentiation of Inverse Trigonometric Functions

This area covers the specific formulas and methods used to differentiate functions like arctan, arcsec, arccsc, and arccot. These functions have well-established derivative formulas which often include expressions such as 1/(1 + y^2) or 1/(|y|?(y^2 - 1)). The derivation of these formulas typically involves the application of implicit differentiation and the chain rule.

Handling Absolute Values in Derivatives

When differentiating inverse trigonometric functions such as arcsec and arccsc, absolute value signs appear in the derivative formulas to ensure that the resulting expressions are valid over the respective domains of the functions. This concept addresses the importance of correctly handling the absolute value to maintain the correctness of the derivative, especially when the function is defined over a domain that includes both positive and negative values.

Domain Considerations

Differentiation of inverse trigonometric functions must take into account the constraints on the domains of these functions. For instance, arcsec and arccsc are only defined for certain ranges of the input variable. Domain considerations ensure that the derivatives are computed correctly and that the formulas are applied in the proper context, preserving the integrity and validity of the mathematical operations.

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