Continuity and Differentiability

$f is continuous at x=a ⟺ lim_{x→a} f(x) = f(a)$

Equivalently, the $left-hand$ limit, $right-hand$ limit, and the function value at a must all be equal: LHL $=$ RHL $= f(a).$

$f, g continuous at a ⟹ f±g, f·g, kf, |f| continuous at a; \frac{f}{g} continuous at a if g(a)≠0$

Continuity is preserved under the usual arithmetic operations, directly inherited from the algebra of limits.

$If g continuous at a and f continuous at g(a), then f∘g is continuous at a$

Used to quickly establish continuity of functions like $sin(x²), e^(cos x),$ etc., built from elementary continuous pieces.

$f is differentiable at x=a ⟺ Lf'(a) = Rf'(a), both finite, where Lf'(a)=lim_{h→0⁻}[f(a+h)−f(a)]/h, Rf'(a)=lim_{h→0⁺}[f(a+h)−f(a)]/h$

The $left-hand$ derivative and $right-hand$ derivative must exist and be equal for f'(a) to exist.

$If f is differentiable at a, then f is continuous at a (converse is FALSE)$

A cornerstone theorem: differentiability is a strictly stronger condition than continuity.

$dy/dx = (dy/du)·(du/dx)$

For a composite function $y = f(u)$ where $u = g(x);$ differentiate the outer function with respect to the inner, then multiply by the derivative of the inner.

$Differentiate both sides of F(x,y)=0 w.r.t. x, treating y as a (differentiable) function of x, then solve for dy/dx$

Used when y is not explicitly isolated as $y=f(x), e.g. x²+y²=r²$ or $xy+sin(y)=x.$

$dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0$

When both x and y are given as functions of a third variable (parameter) $t: x = f(t), y = g(t).$

$For y = [u(x)]^{v(x)}: take ln both sides, ln y = v·ln u, then differentiate implicitly$

Essential whenever both the base and exponent involve x, e.g. xˣ, xˢⁱⁿˣ, (sin x)ˣ — power rule alone cannot handle these.

$d/dx(eˣ)=eˣ; d/dx(aˣ)=aˣ ln a; d/dx(ln x)=\frac{1}{x}; d/dx(log_a x)=1/(x ln a)$

$d/dx(sin⁻¹x)=1/\sqrt{1−x²}; d/dx(cos⁻¹x)=−1/\sqrt{1−x²}; d/dx(tan⁻¹x)=1/(1+x²); d/dx(cot⁻¹x)=−1/(1+x²); d/dx(sec⁻¹x)=1/(|x|\sqrt{x²−1}); d/dx(cosec⁻¹x)=−1/(|x|\sqrt{x²−1})$

Derived via implicit differentiation of the corresponding direct trig function.

$f''(x) = d²y/dx² = d/dx(dy/dx)$

The derivative of the derivative; measures the rate of change of the slope, i.e. concavity of the curve.

$If f continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃ c∈(a,b): f'(c)=0$

Geometrically: if a curve starts and ends at the same height, somewhere in between its tangent must be horizontal.

$If f continuous on [a,b] and differentiable on (a,b), then ∃ c∈(a,b): f'(c) = [f(b)−f(a)]/(b−a)$

The instantaneous rate of change equals the average rate of change somewhere in (a,b); Rolle's theorem is the special case $f(a)=f(b).$