a) Write the derivative of cosech^ 1(x) . [1] b) Define L Hospital's rule. [1] c) Write the condition where the curve y=f(x) has tangent parallel to y axis. [1] d) Write the integral of 1x^2 a^2\,dx…

a) The derivative of ${\text{cosech}}^{-1}(x)$ is given by: $$\frac{d}{dx}\left({\text{cosech}}^{-1}(x)\right)=-\frac{1}{|x|\sqrt{1+x^2}}.$$

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b) L-Hospital's Rule: If ${\lim }_{x\to a}\frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty }{\infty }$, then $$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)},$$ provided the limit on the right exists.

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c) The curve $y=f(x)$ has a tangent parallel to the $y$-axis when its derivative $\frac{dy}{dx}$ is undefined (i.e., when $\frac{dx}{dy}=0$). This occurs when: $$\frac{dx}{dy}=0\text{or}\frac{dy}{dx}\to \infty .$$

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d) The integral of $\int \frac{1}{x^2-a^2}dx$ is evaluated as follows: $$\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln \left|\frac{x-a}{x+a}\right|+C.$$

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e) The standard form of a first-order linear differential equation is: $$\frac{dy}{dx}+P(x)y=Q(x),$$ where $P(x)$ and $Q(x)$ are functions of $x$.