Calculus: Derivative Definitions and Rules

Definition of a Derivative

• The derivative of a function represents the rate at which the function changes as its input changes.

• The fundamental definition of a derivative is given by:
  f^{\prime}(x)={lim}_{h\to0}\frac{f(x+h) - f(x)}{h}

Numerical Derivative

• A numerical derivative can be defined at a specific point $a$ as:
  f^{\prime}(a)={lim}_{h\to0}\frac{f(a+h) - f(a)}{h}

Rules of Differentiation

• The basic rules of differentiation for functions involving products, quotients, and sums are crucial for calculating the derivatives of more complex functions.

Product Rule

• If $u$ and $v$ are functions of $x$, then:
  \frac{d}{dx}(u \cdot v) = u \cdot \frac{d}{dx}(v) + v \cdot \frac{d}{dx}(u)

Quotient Rule

• For functions $u$ and $v$:
  \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \cdot \frac{d}{dx}(u) - u \cdot \frac{d}{dx}(v)}{v^2}

Chain Rule

• For composite functions, the chain rule states:
  \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivatives of Common Functions

• Sine Function (u = sin(x)):

  • \frac{d}{du}[\sin u] = \cos u

• Cosine Function (u = cos(x)):

  • \frac{d}{du}[\cos u] = -\sin u

• Exponential Function (u = e^x):

  • \frac{d}{dx}[e^u] = e^u

• Logarithmic Function (u = log(a) where a is a constant):

  • \frac{d}{du}[\ln u] = \frac{1}{u}

Inverse Trigonometric Functions

• Inverse Sine Function:

  • \frac{d}{du}[\sin^{-1}u] = \frac{1}{\sqrt{1 - u^2}} (for $|u| < 1$)

• Inverse Cosine Function:

  • \frac{d}{du}[\cos^{-1}u] = -\frac{1}{\sqrt{1 - u^2}} (for $|u| < 1$)

• Inverse Tangent Function:

  • \frac{d}{du}[\tan^{-1}u] = \frac{1}{u^2 + 1}

Additional Derivatives

• Cosecant Function:

  • \frac{d}{du}[\csc u] = -\csc u \cdot \cot u

• Secant Function:

  • \frac{d}{du}[\sec u] = \sec u \cdot \tan u

• Cotangent Function:

  • \frac{d}{du}[\cot u] = -\csc^2 u