TL;DR — Quick Summary
Section 1: Introduction to Derivative & Basic Rules
What is a Derivative?
The derivative of a function measures the instantaneous rate of change of the function with respect to its variable. Geometrically, it represents the slope of the tangent line to the curve at any given point.
The derivative of $y = f(x)$ with respect to $x$ is denoted as:
Derivative from First Principles (Definition)
This is the fundamental definition from which all derivative formulas are derived.
Standard Derivatives of Algebraic Functions
| Function $f(x)$ | Derivative $f'(x)$ |
|---|---|
| $c$ (constant) | $0$ |
| $x$ | $1$ |
| $x^n$ | $nx^{n-1}$ |
| $\frac{1}{x}$ | $-\frac{1}{x^2}$ |
| $\sqrt{x}$ | $\frac{1}{2\sqrt{x}}$ |
| $\frac{1}{x^n}$ | $-\frac{n}{x^{n+1}}$ |
Rules of Differentiation
Derivative of Parametric Functions
When $x = f(t)$ and $y = g(t)$, then:
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}$, provided $f'(t) \neq 0$
Derivative of Implicit Functions
For implicit functions where $y$ is not explicitly expressed in terms of $x$ (e.g., $x^2 + y^2 = a^2$), differentiate both sides with respect to $x$ and solve for $\frac{dy}{dx}$.
Example: For $x^2 + y^2 = a^2$, $2x + 2y\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$
Higher Order Derivatives
First derivative: $\frac{dy}{dx}$ or $y_1$ or $f'(x)$
Second derivative: $\frac{d^2y}{dx^2}$ or $y_2$ or $f''(x)$
Third derivative: $\frac{d^3y}{dx^3}$ or $y_3$ or $f'''(x)$
nth derivative: $\frac{d^ny}{dx^n}$ or $y_n$ or $f^{(n)}(x)$
Section 2: Derivatives of Trigonometric & Inverse Trigonometric Functions
Derivatives of Trigonometric Functions
| Function $f(x)$ | Derivative $f'(x)$ |
|---|---|
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $\cot x$ | $-\csc^2 x$ |
| $\sec x$ | $\sec x \tan x$ |
| $\csc x$ | $-\csc x \cot x$ |
Derivatives of Inverse Trigonometric Functions
| Function $f(x)$ | Derivative $f'(x)$ | Domain Restriction |
|---|---|---|
| $\sin^{-1} x$ | $\frac{1}{\sqrt{1-x^2}}$ | $ |
| $\cos^{-1} x$ | $-\frac{1}{\sqrt{1-x^2}}$ | $ |
| $\tan^{-1} x$ | $\frac{1}{1+x^2}$ | All real $x$ |
| $\cot^{-1} x$ | $-\frac{1}{1+x^2}$ | All real $x$ |
| $\sec^{-1} x$ | $\frac{1}{ | x |
| $\csc^{-1} x$ | $-\frac{1}{ | x |
Important Derivatives with Chain Rule
$\frac{d}{dx}[\sin(f(x))] = \cos(f(x)) \cdot f'(x)$
$\frac{d}{dx}[\cos(f(x))] = -\sin(f(x)) \cdot f'(x)$
$\frac{d}{dx}[\tan(f(x))] = \sec^2(f(x)) \cdot f'(x)$
Section 3: Derivatives of Exponential & Logarithmic Functions
Exponential Functions
Natural Exponential: $\frac{d}{dx}(e^x) = e^x$
General Exponential: $\frac{d}{dx}(a^x) = a^x \ln a$, where $a > 0, a \neq 1$
With Chain Rule: $\frac{d}{dx}(e^{f(x)}) = e^{f(x)} \cdot f'(x)$
$\frac{d}{dx}(a^{f(x)}) = a^{f(x)} \ln a \cdot f'(x)$
Logarithmic Functions
Natural Logarithm: $\frac{d}{dx}(\ln x) = \frac{1}{x}$, for $x > 0$
General Logarithm: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$, for $x > 0, a > 0, a \neq 1$
With Chain Rule: $\frac{d}{dx}[\ln(f(x))] = \frac{f'(x)}{f(x)}$, for $f(x) > 0$
$\frac{d}{dx}[\log_a(f(x))] = \frac{f'(x)}{f(x)\ln a}$, for $f(x) > 0$
Special Derivatives
$\frac{d}{dx}(\ln |x|) = \frac{1}{x}$, for $x \neq 0$
$\frac{d}{dx}(x^x) = x^x(1 + \ln x)$ (using logarithmic differentiation)
Key Points to Remember
- Derivative is the slope of the tangent line at any point.
- Chain rule is your best friend—use it whenever you have composite functions.
- For inverse trigonometric functions, always check the domain restrictions.
- Logarithmic differentiation is useful when differentiating complex products, quotients, or functions of the form $[f(x)]^{g(x)}$.
Practice Problems
Q1. Find $\frac{dy}{dx}$ for $y = x^3 - 5x^2 + 7x - 2$
Q2. Differentiate $y = \sin(2x)$ using the chain rule.
Q3. Find $\frac{dy}{dx}$ for $y = \ln(x^2 + 1)$
Q4. For the parametric equations $x = t^2$ and $y = t^3$, find $\frac{dy}{dx}$
Q5. Find $\frac{d^2y}{dx^2}$ for $y = e^{2x}$