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′(x)=limh→0f(x+h)−f(x)hf^{\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′(a)=limh→0f(a+h)−f(a)hf^{\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:
    ddx(u⋅v)=u⋅ddx(v)+v⋅ddx(u)\frac{d}{dx}(u \cdot v) = u \cdot \frac{d}{dx}(v) + v \cdot \frac{d}{dx}(u)

Quotient Rule

  • For functions $u$ and $v$:
    ddx(uv)=v⋅ddx(u)−u⋅ddx(v)v2\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:
    ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivatives of Common Functions

  • Sine Function (u = sin(x)):

    • ddu[sin⁡u]=cos⁡u\frac{d}{du}[\sin u] = \cos u

  • Cosine Function (u = cos(x)):

    • ddu[cos⁡u]=−sin⁡u\frac{d}{du}[\cos u] = -\sin u

  • Exponential Function (u = e^x):

    • ddx[eu]=eu\frac{d}{dx}[e^u] = e^u

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

    • ddu[ln⁡u]=1u\frac{d}{du}[\ln u] = \frac{1}{u}

Inverse Trigonometric Functions

  • Inverse Sine Function:

    • ddu[sin⁡−1u]=11−u2\frac{d}{du}[\sin^{-1}u] = \frac{1}{\sqrt{1 - u^2}} (for $|u| < 1$)

  • Inverse Cosine Function:

    • ddu[cos⁡−1u]=−11−u2\frac{d}{du}[\cos^{-1}u] = -\frac{1}{\sqrt{1 - u^2}} (for $|u| < 1$)

  • Inverse Tangent Function:

    • ddu[tan⁡−1u]=1u2+1\frac{d}{du}[\tan^{-1}u] = \frac{1}{u^2 + 1}

Additional Derivatives

  • Cosecant Function:

    • ddu[csc⁡u]=−csc⁡u⋅cot⁡u\frac{d}{du}[\csc u] = -\csc u \cdot \cot u

  • Secant Function:

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

  • Cotangent Function:

    • ddu[cot⁡u]=−csc⁡2u\frac{d}{du}[\cot u] = -\csc^2 u