Continuity and Differentiability Mind Map

A function f is continuous at x = a if f(a) is defined, lim(x -> a) f(x) exists, and lim(x -> a) f(x) = f(a).

At a joining point, first decide which expression gives f(a), then compare it with the left-hand and right-hand limits.

A function f is differentiable at x = a if its left-hand derivative and right-hand derivative at a both exist and are equal.

Before testing differentiability of a piecewise function, first check continuity. If continuity fails, differentiability fails immediately.

The chain rule gives the derivative of a composite function f(g(x)) by differentiating the outer function at g(x) and multiplying by the derivative of g(x).

Circle the inner function mentally before differentiating; the final derivative must contain the derivative of that inner part.

Implicit differentiation is used when x and y are related by an equation and y is not first written explicitly as a function of x.

Whenever differentiating a y-term with respect to x, attach dy/dx unless the term is a constant.

This concept covers standard derivatives of inverse trigonometric functions, exponential functions, and logarithmic functions, along with their domain conditions.

For inverse trigonometric functions, write the formula first, then substitute the inner expression and multiply by its derivative.

Logarithmic differentiation is a method where logarithms are taken on both sides before differentiating, especially for variable powers and complicated products or quotients.

After differentiating log y, always write dy/dx = y times the right-hand side, then substitute the original value of y.

For parametric equations x = f(t), y = g(t), the derivative dy/dx is found as (dy/dt)/(dx/dt), provided dx/dt is not zero. The second-order derivative measures the rate of change of dy/dx with respect to x.

For second derivative in parametric form, use d2y/dx2 = [d/dt(dy/dx)]/(dx/dt).