Calculus - Derivatives of Composite, Implicit, Inverse Trigonometric, Exponential and Logarithmic Functions

Chain Rule for Composite Functions: When a function is nested within another, such as $y = f(g(x))$, the derivative is calculated as the derivative of the outer function multiplied by the derivative of the inner function, expressed as $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$. Visually, this process is like peeling an onion; you differentiate the outermost layer first and work your way inward. On a graph, the steepness of the composite curve is determined by the compounding rates of change of its individual components.

Implicit Differentiation: This technique is used for equations where $y$ is not explicitly isolated, such as $x^2 + y^2 = r^2$. We differentiate every term with respect to $x$, and whenever we differentiate a term involving $y$, we multiply by $\frac{dy}{dx}$ as per the chain rule. Geometrically, implicit differentiation allows us to find the slope of the tangent line to curves that may not be functions (like circles or ellipses) at any point $(x, y)$ on the curve.

Derivatives of Exponential Functions: The natural exponential function $e^x$ is unique because its slope at any point is exactly equal to its y-value. Visually, the graph of $y = e^x$ becomes steeper as it moves to the right, and the rate of this steepening is the function itself. For a general base $a$, the derivative of $a^x$ is $a^x \ln a$, where the $\ln a$ factor acts as a scaling constant for the growth rate.

Logarithmic Differentiation: This is a powerful method used for complex products, quotients, or functions of the form $y = [f(x)]^{g(x)}$. By taking the natural logarithm $(\ln)$ of both sides, we use log properties to turn powers into products and products into sums. After differentiating implicitly, we solve for $\frac{dy}{dx}$. This method linearizes the relationship between the base and the exponent, making the differentiation of variable-power functions manageable.

Derivatives of Inverse Trigonometric Functions: Functions like $\sin^{-1} x$ and $\tan^{-1} x$ have derivatives that are algebraic fractions. For example, the derivative of $\tan^{-1} x$ is $\frac{1}{1+x^2}$. Visually, the inverse tangent function has horizontal asymptotes at $y = \pm \frac{\pi}{2}$ and is strictly increasing; its derivative represents a bell-shaped curve that peaks at $x=0$, reflecting the steepest part of the original function's graph.

Logarithmic Functions: The derivative of $\ln x$ is $\frac{1}{x}$ for $x > 0$. Visually, the natural log curve grows slower and slower as $x$ increases, which is reflected in its derivative $\frac{1}{x}$ approaching zero. If the base of the log is not $e$, we use the change of base formula, resulting in $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$.