1. **Problem Statement:** We will explore key concepts in calculus including continuity, differentiability, chain rule, derivatives of inverse trigonometric functions, implicit differentiation, exponential and logarithmic functions, logarithmic differentiation, parametric derivatives, and second order derivatives. 2. **Continuity and Differentiability:** A function $f(x)$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$. It is differentiable at $x=a$ if the derivative $f'(a)$ exists, meaning the function has a defined tangent slope there. 3. **Derivative of Composite Functions (Chain Rule):** For $y = f(g(x))$, the derivative is given by $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x).$$ This means differentiate the outer function evaluated at the inner function times the derivative of the inner function. 4. **Derivatives of Inverse Trigonometric Functions:** For example, $$\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1-x^2}},$$ valid for $|x|<1$. Similar formulas exist for $\cos^{-1} x$, $\tan^{-1} x$, etc. 5. **Implicit Differentiation:** When $y$ is defined implicitly by an equation involving $x$ and $y$, differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (using $\frac{dy}{dx}$), then solve for $\frac{dy}{dx}$. 6. **Exponential and Logarithmic Functions:** The derivative of $e^x$ is $e^x$. For $a^x$ where $a>0$, $$\frac{d}{dx} a^x = a^x \ln a.$$ The derivative of $\ln x$ is $\frac{1}{x}$. 7. **Logarithmic Differentiation:** Useful for functions like $y = f(x)^{g(x)}$. Take natural log on both sides, differentiate implicitly, then solve for $\frac{dy}{dx}$. 8. **Parametric Derivatives:** If $x = x(t)$ and $y = y(t)$, then $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ provided $\frac{dx}{dt} \neq 0$. 9. **Second Order Derivatives:** The second derivative is the derivative of the first derivative, denoted $$\frac{d^2 y}{dx^2} = \frac{d}{dx} \left( \frac{dy}{dx} \right).$$ It gives information about the concavity of the function. This overview covers the fundamental differentiation techniques and concepts requested.