Derivatives and Differentiation Rules

• A derivative of a function is its instantaneous rate of change at a point.
• The derivative of $f(x)$ at $x=a$ is defined as $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$, if the limit exists.
• If the limit exists, the function is differentiable at $x = a$.
• Geometrically, $f'(a)$ is the slope of the tangent line to the graph of $f(x)$ at the point $(a, f(a))$.

Lateral Derivatives

• The derivative from the right is given by $f'(a^+) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$.
• The derivative from the left is given by $f'(a^-) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}$.
• A function is differentiable at $x = a$ if and only if $f'(a^+) = f'(a^-)$.
• The derivative function $f'(x)$ is defined as $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.

Rules of Derivation

• Constant Rule: If $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$.
• Power Rule: If $f(x) = x^n$, where $n$ is a real number, then $f'(x) = nx^{n-1}$.
• Sum or Difference Rule: If $f(x) = u(x) \pm v(x)$, then $f'(x) = u'(x) \pm v'(x)$.
• Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.
• Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$, where $v(x) \neq 0$.
• Chain Rule: If $f(x) = u(v(x))$, then $f'(x) = u'(v(x)) \cdot v'(x)$.

Derivatives of Trigonometric Functions

• $(\sin x)' = \cos x$
• $(\cos x)' = -\sin x$
• $(\tan x)' = \sec^2 x$
• $(\cot x)' = -\csc^2 x$
• $(\sec x)' = \sec x \tan x$
• $(\csc x)' = -\csc x \cot x$

Derivatives of Exponential and Logarithmic Functions

• $(e^x)' = e^x$
• $(a^x)' = a^x \ln a$
• $(\ln x)' = \frac{1}{x}$
• $(\log_a x)' = \frac{1}{x \ln a}$

Implicit Differentiation

Process

• Differentiate both sides of the equation with respect to x.
• Apply the chain rule when necessary.
• Solve for $\frac{dy}{dx}$.

Applications of the Derivative

Tangent and Normal Lines

• Tangent Line: $y - f(a) = f'(a)(x - a)$
• Normal Line: $y - f(a) = -\frac{1}{f'(a)}(x - a)$, if $f'(a) \neq 0$

Maximum and Minimum Values

• Critical Points: Where $f'(x) = 0$ or $f'(x)$ does not exist.
• First Derivative Test: Determines if a critical point is a local max or min.
• Second Derivative Test: If $f''(c) > 0$, then $f(c)$ is a local minimum. If $f''(c) < 0$, then $f(c)$ is a local maximum.

Rolle's Theorem

• If $f(x)$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists at least one point $c$ in $(a, b)$ such that $f'(c) = 0$.

Mean Value Theorem

• If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one point $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

L'Hopital's Rule

• If $\lim_{x \to c} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$, provided this limit exists.