Differentiation Cheat Sheet

Definition of Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. Mathematically, the derivative of a function \( f(x) \) is defined as:

\[ f'(x) = \lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h} \]

This definition expresses the instantaneous rate of change of \( f(x) \) at a given point \( x \).

Instantaneous Rate of Change

The derivative \( f'(x) \) represents the slope of the tangent line to the curve at \( x \).

Formulas and Properties of Differentiation

1. Basic Derivatives

\[ \frac{d}{dx} (c) = 0 \quad \text{(where \( c \) is a constant)} \] \[ \frac{d}{dx} (x) = 1 \] \[ \frac{d}{dx} (x^n) = n x^{n-1}, \quad n \in \mathbb{R} \] \[ \frac{d}{dx} (\sqrt{x}) = \frac{1}{2\sqrt{x}} \] \[ \frac{d}{dx} \left(\frac{1}{x}\right) = -\frac{1}{x^2} \]

2. Trigonometric Derivatives

\[ \frac{d}{dx} (\sin x) = \cos x \] \[ \frac{d}{dx} (\cos x) = -\sin x \] \[ \frac{d}{dx} (\tan x) = \sec^2 x, \quad x \neq \frac{\pi}{2} + n\pi \] \[ \frac{d}{dx} (\cot x) = -\csc^2 x, \quad x \neq n\pi \]

3. Logarithmic and Exponential Derivatives

\[ \frac{d}{dx} (\ln x) = \frac{1}{x}, \quad x > 0 \] \[ \frac{d}{dx} (e^x) = e^x \] \[ \frac{d}{dx} (a^x) = a^x \ln a, \quad a > 0 \]

4. Differentiation Rules

(i) Sum and Difference Rule

\[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \]

(ii) Product Rule

\[ \frac{d}{dx} [u(x) v(x)] = u’ v + u v’ \]

(iii) Quotient Rule

\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u’ v – u v’}{v^2}, \quad v(x) \neq 0 \]

(iv) Chain Rule

If \( y = f(g(x)) \), then: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

Examples of Differentiation

Example 1: Derivative of a Polynomial Function

Let \( f(x) = x^2 + 3x + 5 \). Using the first principles: \[ f'(x) = \lim\limits_{h \to 0} \frac{(x+h)^2 + 3(x+h) + 5 – (x^2 + 3x + 5)}{h} \] Expanding: \[ = \lim\limits_{h \to 0} \frac{x^2 + 2xh + h^2 + 3x + 3h + 5 – x^2 – 3x – 5}{h} \] \[ = \lim\limits_{h \to 0} \frac{2xh + h^2 + 3h}{h} \] \[ = \lim\limits_{h \to 0} (2x + h + 3) \] \[ = 2x + 3 \] Thus, \[ \frac{d}{dx}(x^2 + 3x + 5) = 2x + 3 \]

Example 2: Derivative of Trigonometric Functions

Let \( f(x) = \sin x \), then using first principles: \[ f'(x) = \lim\limits_{h \to 0} \frac{\sin(x+h) – \sin x}{h} \] Using the identity \( \sin(A+B) = \sin A \cos B + \cos A \sin B \), \[ = \lim\limits_{h \to 0} \frac{\sin x \cos h + \cos x \sin h – \sin x}{h} \] \[ = \lim\limits_{h \to 0} \frac{\sin x (\cos h -1) + \cos x \sin h}{h} \] Since \( \lim\limits_{h \to 0} \frac{\sin h}{h} = 1 \) and \( \lim\limits_{h \to 0} \frac{\cos h -1}{h} = 0 \), \[ = \cos x \] Thus, \[ \frac{d}{dx}(\sin x) = \cos x \]

Applications of Differentiation

• Finding Tangents and Normals: The slope of the tangent line at \( x \) is \( f'(x) \), and the normal’s slope is \( -\frac{1}{f'(x)} \).
• Maxima and Minima: If \( f'(x) = 0 \) and \( f”(x) > 0 \), it’s a local minimum; if \( f”(x) < 0 \), it's a local maximum.
• Rate of Change: \( f'(x) \) gives the instantaneous rate of change of a quantity.