a) Write the slope of tangent to the curve y = f(x) at (x 1, y 1) . [1] b) Write the derivative of cosech x with respect to x . [1] c) A differential equation is in the form dydx + Py = Q , where P…

a) The slope of the tangent to the curve $y=f(x)$ at the point $(x_1,y_1)$ is given by the derivative of $f(x)$ evaluated at $x_1$. Mathematically, it is expressed as: $$\boxed{f^{\prime }(x_1)}$$

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b) The derivative of $\mathrm{cosech}x$ with respect to $x$ is: $$\frac{d}{dx}(\mathrm{cosech}x)=-\mathrm{cosech}x\cot x$$ $$\boxed{-\mathrm{cosech}x\cot x}$$

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c) The given differential equation $\frac{dy}{dx}+Py=Q$ is known as a first-order linear differential equation.

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d) The integral of $\int \frac{1}{a^2-x^2}dx$ can be evaluated using partial fractions. The result is: $$\int \frac{1}{a^2-x^2}dx=\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C$$ $$\boxed{\frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right|+C}$$

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e) One characteristic of L'Hôpital's Rule is that it can be applied to indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty }{\infty }$ when evaluating limits. Specifically, if ${\lim }_{x\to c}\frac{f(x)}{g(x)}$ is of an indeterminate form, then: $$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\underset{x\to c}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$$ provided the limit on the right exists. The rule can be applied repeatedly if the resulting limit is still indeterminate.